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L3 Lab 3

case bound E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r x✝ : ℝ hx : x✝ ∈ Icc a b r : ℝ hr : r > 0 x : ℝ a✝¹ : x ∈ Ico a b a✝ : f x = B x + r * (x - a) ⊢ B' x < B' x + r

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one]

exact (lt_add_iff_pos_right _).2 hr

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one]

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

case a E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r x : ℝ hx : x ∈ Icc a b r : ℝ hr : r > 0 ⊢ x ∈ Icc a b

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr

exact hx

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) ⊢ ∀ ⦃x : ℝ⦄, x ∈ Icc a b → f x ≤ B x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx

intro x hx

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) x : ℝ hx : x ∈ Icc a b ⊢ f x ≤ B x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx

have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) x : ℝ hx : x ∈ Icc a b this : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 ⊢ f x ≤ B x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)

convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

case h.e'_4 E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) x : ℝ hx : x ∈ Icc a b this : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 ⊢ B x = B x + 0 * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;>

simp

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;>

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → ℝ a b : ℝ hf : ContinuousOn f (Icc a b) B B' : ℝ → ℝ ha : f a ≤ B a hB : ContinuousOn B (Icc a b) hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x bound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) x : ℝ hx : x ∈ Icc a b this : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 ⊢ 0 ∈ closure (Ioi 0)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;>

simp

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;>

Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I

/-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

let g x := f x - f a

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a

have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const

have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) ⊢ ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by

intro x hx

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) x : ℝ hx : x ∈ Ico a b ⊢ HasDerivWithinAt g (f' x) (Ici x) x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx

simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)

let B x := C * (x - a)

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a)

have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a)

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) ⊢ ∀ (x : ℝ), HasDerivAt B C x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by

intro x

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) x : ℝ ⊢ HasDerivAt B C x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x

simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) hB : ∀ (x : ℝ), HasDerivAt B C x ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))

convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) hB : ∀ (x : ℝ), HasDerivAt B C x ⊢ ‖g a‖ ≤ B a

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound

simp only

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ContinuousOn f (Icc a b) hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C g : ℝ → E := fun x => f x - f a hg : ContinuousOn g (Icc a b) hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x B : ℝ → ℝ := fun x => C * (x - a) hB : ∀ (x : ℝ), HasDerivAt B C x ⊢ ‖f a - f a‖ ≤ C * (a - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only;

rw [sub_self, norm_zero, sub_self, mul_zero]

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only;

Mathlib.Analysis.Calculus.MeanValue.337_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

Mathlib.Analysis.Calculus.MeanValue.355_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C x : ℝ hx : x ∈ Ico a b ⊢ HasDerivWithinAt (fun x => f x) (f' x) (Ici x) x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound

exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx)

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound

Mathlib.Analysis.Calculus.MeanValue.355_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b C : ℝ hf : DifferentiableOn ℝ f (Icc a b) bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C ⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by

Mathlib.Analysis.Calculus.MeanValue.367_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b C : ℝ hf : DifferentiableOn ℝ f (Icc a b) bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C ⊢ ∀ x ∈ Icc a b, HasDerivWithinAt (fun x => f x) (derivWithin f (Icc a b) x) (Icc a b) x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound

exact fun x hx => (hf x hx).hasDerivWithinAt

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound

Mathlib.Analysis.Calculus.MeanValue.367_0.ReDurB0qNQAwk9I

/-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' : ℝ → E C : ℝ hf : ∀ x ∈ Icc 0 1, HasDerivWithinAt f (f' x) (Icc 0 1) x bound : ∀ x ∈ Ico 0 1, ‖f' x‖ ≤ C ⊢ ‖f 1 - f 0‖ ≤ C

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by

simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)

/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by

Mathlib.Analysis.Calculus.MeanValue.377_0.ReDurB0qNQAwk9I

/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b C : ℝ hf : DifferentiableOn ℝ f (Icc 0 1) bound : ∀ x ∈ Ico 0 1, ‖derivWithin f (Icc 0 1) x‖ ≤ C ⊢ ‖f 1 - f 0‖ ≤ C

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by

simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one)

/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by

Mathlib.Analysis.Calculus.MeanValue.387_0.ReDurB0qNQAwk9I

/-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ hcont : ContinuousOn f (Icc a b) hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x ⊢ ∀ x ∈ Icc a b, f x = f a

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by

have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx

theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by

Mathlib.Analysis.Calculus.MeanValue.396_0.ReDurB0qNQAwk9I

theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ hcont : ContinuousOn f (Icc a b) hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x this : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) ⊢ ∀ x ∈ Icc a b, f x = f a

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx

simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this

theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx

Mathlib.Analysis.Calculus.MeanValue.396_0.ReDurB0qNQAwk9I

theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ hdiff : DifferentiableOn ℝ f (Icc a b) hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0 ⊢ ∀ x ∈ Icc a b, f x = f a

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by

have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by

Mathlib.Analysis.Calculus.MeanValue.403_0.ReDurB0qNQAwk9I

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ hdiff : DifferentiableOn ℝ f (Icc a b) hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0 ⊢ ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by

simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by

Mathlib.Analysis.Calculus.MeanValue.403_0.ReDurB0qNQAwk9I

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ hdiff : DifferentiableOn ℝ f (Icc a b) hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0 H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 ⊢ ∀ x ∈ Icc a b, f x = f a

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx

simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx

Mathlib.Analysis.Calculus.MeanValue.403_0.ReDurB0qNQAwk9I

theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x fcont : ContinuousOn f (Icc a b) gcont : ContinuousOn g (Icc a b) hi : f a = g a ⊢ ∀ y ∈ Icc a b, f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by

simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by

Mathlib.Analysis.Calculus.MeanValue.413_0.ReDurB0qNQAwk9I

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x fcont : ContinuousOn f (Icc a b) gcont : ContinuousOn g (Icc a b) hi : f a - g a = 0 ⊢ ∀ y ∈ Icc a b, f y - g y = 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢

exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy)

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢

Mathlib.Analysis.Calculus.MeanValue.413_0.ReDurB0qNQAwk9I

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x fcont : ContinuousOn f (Icc a b) gcont : ContinuousOn g (Icc a b) hi : f a - g a = 0 y : ℝ hy : y ∈ Ico a b ⊢ HasDerivWithinAt (fun y => f y - g y) 0 (Ici y) y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by

simpa only [sub_self] using (derivf y hy).sub (derivg y hy)

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by

Mathlib.Analysis.Calculus.MeanValue.413_0.ReDurB0qNQAwk9I

/-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E fdiff : DifferentiableOn ℝ f (Icc a b) gdiff : DifferentiableOn ℝ g (Icc a b) hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b) hi : f a = g a ⊢ ∀ y ∈ Icc a b, f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by

have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by

Mathlib.Analysis.Calculus.MeanValue.423_0.ReDurB0qNQAwk9I

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E fdiff : DifferentiableOn ℝ f (Icc a b) gdiff : DifferentiableOn ℝ g (Icc a b) hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b) hi : f a = g a A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y ⊢ ∀ y ∈ Icc a b, f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

Mathlib.Analysis.Calculus.MeanValue.423_0.ReDurB0qNQAwk9I

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f : ℝ → E a b : ℝ f' g : ℝ → E fdiff : DifferentiableOn ℝ f (Icc a b) gdiff : DifferentiableOn ℝ g (Icc a b) hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b) hi : f a = g a A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y ⊢ ∀ y ∈ Icc a b, f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy)

Mathlib.Analysis.Calculus.MeanValue.423_0.ReDurB0qNQAwk9I

/-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by

letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/

set g := (AffineMap.lineMap x y : ℝ → E)

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G g : ℝ → E := ⇑(AffineMap.lineMap x y) ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E)

have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E)

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G g : ℝ → E := ⇑(AffineMap.lineMap x y) segm : MapsTo g (Icc 0 1) s ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys

have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G g : ℝ → E := ⇑(AffineMap.lineMap x y) segm : MapsTo g (Icc 0 1) s t : ℝ ht : t ∈ Icc 0 1 ⊢ HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by

simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G g : ℝ → E := ⇑(AffineMap.lineMap x y) segm : MapsTo g (Icc 0 1) s hD : ∀ t ∈ Icc 0 1, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm

have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound✝ : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s this : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G g : ℝ → E := ⇑(AffineMap.lineMap x y) segm : MapsTo g (Icc 0 1) s hD : ∀ t ∈ Icc 0 1, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) (Icc 0 1) t bound : ∀ t ∈ Ico 0 1, ‖(f' (g t)) (y - x)‖ ≤ C * ‖y - x‖ ⊢ ‖f y - f x‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _

simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _

Mathlib.Analysis.Calculus.MeanValue.455_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C✝ : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G C : ℝ≥0 hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C hs : Convex ℝ s ⊢ LipschitzOnWith C f s

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by

rw [lipschitzOnWith_iff_norm_sub_le]

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by

Mathlib.Analysis.Calculus.MeanValue.476_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C✝ : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G C : ℝ≥0 hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C hs : Convex ℝ s ⊢ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x - f y‖ ≤ ↑C * ‖x - y‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le]

intro x x_in y y_in

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le]

Mathlib.Analysis.Calculus.MeanValue.476_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C✝ : ℝ s : Set E x✝ y✝ : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G C : ℝ≥0 hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C hs : Convex ℝ s x : E x_in : x ∈ s y : E y_in : y ∈ s ⊢ ‖f x - f y‖ ≤ ↑C * ‖x - y‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in

exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in

Mathlib.Analysis.Calculus.MeanValue.476_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by

obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by

Mathlib.Analysis.Calculus.MeanValue.487_0.ReDurB0qNQAwk9I

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ⊢ ∃ ε > 0, ball x ε ∩ s ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K} case intro.intro E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ε : ℝ ε0 : ε > 0 hε : ball x ε ∩ s ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K} ⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }

exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K }

Mathlib.Analysis.Calculus.MeanValue.487_0.ReDurB0qNQAwk9I

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Mathlib_Analysis_Calculus_MeanValue

case intro.intro E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ε : ℝ ε0 : ε > 0 hε : ball x ε ∩ s ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K} ⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))

rw [inter_comm] at hε

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))

Mathlib.Analysis.Calculus.MeanValue.487_0.ReDurB0qNQAwk9I

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Mathlib_Analysis_Calculus_MeanValue

case intro.intro E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ε : ℝ ε0 : ε > 0 hε : s ∩ ball x ε ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K} ⊢ ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε

refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε

Mathlib.Analysis.Calculus.MeanValue.487_0.ReDurB0qNQAwk9I

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Mathlib_Analysis_Calculus_MeanValue

case intro.intro E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f✝ g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s f : E → G hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y hcont : ContinuousWithinAt f' s x K : ℝ≥0 hK : ‖f' x‖₊ < K ε : ℝ ε0 : ε > 0 hε : s ∩ ball x ε ⊆ {y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K} ⊢ LipschitzOnWith K f (s ∩ ball x ε)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩

exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩

Mathlib.Analysis.Calculus.MeanValue.487_0.ReDurB0qNQAwk9I

/-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C✝ : ℝ s : Set E✝ x y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G C : ℝ≥0 hf : Differentiable 𝕜 f bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C ⊢ LipschitzWith C f

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by

let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by

Mathlib.Analysis.Calculus.MeanValue.553_0.ReDurB0qNQAwk9I

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C✝ : ℝ s : Set E✝ x y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G C : ℝ≥0 hf : Differentiable 𝕜 f bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E ⊢ LipschitzWith C f

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

rw [← lipschitzOn_univ]

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

Mathlib.Analysis.Calculus.MeanValue.553_0.ReDurB0qNQAwk9I

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C✝ : ℝ s : Set E✝ x y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G C : ℝ≥0 hf : Differentiable 𝕜 f bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E ⊢ LipschitzOnWith C f univ

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ]

exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ]

Mathlib.Analysis.Calculus.MeanValue.553_0.ReDurB0qNQAwk9I

/-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s ⊢ ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/

let g y := f y - φ y

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y ⊢ ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y

have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt

calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by

simp

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ ‖f y - f x - (φ y - φ x)‖ = ‖f y - φ y - (f x - φ x)‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by

congr 1

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

case e_a E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ f y - f x - (φ y - φ x) = f y - φ y - (f x - φ x)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1;

abel

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1;

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

case e_a E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ f y - f x - (φ y - φ x) = f y - φ y - (f x - φ x)

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1;

abel

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1;

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g✝ : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C hs : Convex ℝ s xs : x ∈ s ys : y ∈ s g : E → G := fun y => f y - φ y hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x ⊢ ‖f y - φ y - (f x - φ x)‖ = ‖g y - g x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by

simp

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by

Mathlib.Analysis.Calculus.MeanValue.563_0.ReDurB0qNQAwk9I

/-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0 hx : x ∈ s hy : y ∈ s ⊢ f x = f y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by

have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl]

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by

Mathlib.Analysis.Calculus.MeanValue.598_0.ReDurB0qNQAwk9I

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x✝ y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0 hx✝ : x✝ ∈ s hy : y ∈ s x : E hx : x ∈ s ⊢ ‖fderivWithin 𝕜 f s x‖ ≤ 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by

simp only [hf' x hx, norm_zero, le_rfl]

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by

Mathlib.Analysis.Calculus.MeanValue.598_0.ReDurB0qNQAwk9I

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0 hx : x ∈ s hy : y ∈ s bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 ⊢ f x = f y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl]

simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl]

Mathlib.Analysis.Calculus.MeanValue.598_0.ReDurB0qNQAwk9I

/-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C : ℝ s : Set E✝ x✝ y✝ : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G hf : Differentiable 𝕜 f hf' : ∀ (x : E), fderiv 𝕜 f x = 0 x y : E ⊢ f x = f y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by

let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by

Mathlib.Analysis.Calculus.MeanValue.608_0.ReDurB0qNQAwk9I

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C : ℝ s : Set E✝ x✝ y✝ : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G hf : Differentiable 𝕜 f hf' : ∀ (x : E), fderiv 𝕜 f x = 0 x y : E A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E ⊢ f x = f y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

Mathlib.Analysis.Calculus.MeanValue.608_0.ReDurB0qNQAwk9I

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C : ℝ s : Set E✝ x✝² y✝ : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G hf : Differentiable 𝕜 f hf' : ∀ (x : E), fderiv 𝕜 f x = 0 x✝¹ y : E A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E x : E x✝ : x ∈ univ ⊢ fderivWithin 𝕜 f univ x = 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by

rw [fderivWithin_univ]

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by

Mathlib.Analysis.Calculus.MeanValue.608_0.ReDurB0qNQAwk9I

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g : E✝ → G C : ℝ s : Set E✝ x✝² y✝ : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f : E → G hf : Differentiable 𝕜 f hf' : ∀ (x : E), fderiv 𝕜 f x = 0 x✝¹ y : E A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E x : E x✝ : x ∈ univ ⊢ fderiv 𝕜 f x = 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ];

exact hf' x

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ];

Mathlib.Analysis.Calculus.MeanValue.608_0.ReDurB0qNQAwk9I

theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y✝ : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hg : DifferentiableOn 𝕜 g s hs' : UniqueDiffOn 𝕜 s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x hx : x ∈ s hfgx : f x = g x y : E hy : y ∈ s ⊢ f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by

suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by

Mathlib.Analysis.Calculus.MeanValue.617_0.ReDurB0qNQAwk9I

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y✝ : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hg : DifferentiableOn 𝕜 g s hs' : UniqueDiffOn 𝕜 s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x hx : x ∈ s hfgx : f x = g x y : E hy : y ∈ s this : f x - g x = f y - g y ⊢ f y = g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by

rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by

Mathlib.Analysis.Calculus.MeanValue.617_0.ReDurB0qNQAwk9I

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y✝ : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hg : DifferentiableOn 𝕜 g s hs' : UniqueDiffOn 𝕜 s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x hx : x ∈ s hfgx : f x = g x y : E hy : y ∈ s ⊢ f x - g x = f y - g y

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this

refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this

Mathlib.Analysis.Calculus.MeanValue.617_0.ReDurB0qNQAwk9I

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f g : E → G C : ℝ s : Set E x y✝ : E f' g' : E → E →L[𝕜] G φ : E →L[𝕜] G hs : Convex ℝ s hf : DifferentiableOn 𝕜 f s hg : DifferentiableOn 𝕜 g s hs' : UniqueDiffOn 𝕜 s hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x hx : x ∈ s hfgx : f x = g x y : E hy : y ∈ s z : E hz : z ∈ s ⊢ fderivWithin 𝕜 (fun y => f y - g y) s z = 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy

rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy

Mathlib.Analysis.Calculus.MeanValue.617_0.ReDurB0qNQAwk9I

/-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g✝ : E✝ → G C : ℝ s : Set E✝ x✝ y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f g : E → G hf : Differentiable 𝕜 f hg : Differentiable 𝕜 g hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x x : E hfgx : f x = g x ⊢ f = g

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by

let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by

Mathlib.Analysis.Calculus.MeanValue.628_0.ReDurB0qNQAwk9I

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g✝ : E✝ → G C : ℝ s : Set E✝ x✝ y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f g : E → G hf : Differentiable 𝕜 f hg : Differentiable 𝕜 g hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x x : E hfgx : f x = g x A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E ⊢ f = g

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E

Mathlib.Analysis.Calculus.MeanValue.628_0.ReDurB0qNQAwk9I

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g✝ : E✝ → G C : ℝ s : Set E✝ x✝ y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f g : E → G hf : Differentiable 𝕜 f hg : Differentiable 𝕜 g hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x x : E hfgx : f x = g x A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E ⊢ EqOn f g univ

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x

exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x

Mathlib.Analysis.Calculus.MeanValue.628_0.ReDurB0qNQAwk9I

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g

Mathlib_Analysis_Calculus_MeanValue

E✝ : Type u_1 inst✝⁹ : NormedAddCommGroup E✝ inst✝⁸ : NormedSpace ℝ E✝ F : Type u_2 inst✝⁷ : NormedAddCommGroup F inst✝⁶ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝⁵ : IsROrC 𝕜 inst✝⁴ : NormedSpace 𝕜 E✝ inst✝³ : NormedAddCommGroup G inst✝² : NormedSpace 𝕜 G f✝ g✝ : E✝ → G C : ℝ s : Set E✝ x✝² y : E✝ f' g' : E✝ → E✝ →L[𝕜] G φ : E✝ →L[𝕜] G E : Type u_5 inst✝¹ : NormedAddCommGroup E inst✝ : NormedSpace 𝕜 E f g : E → G hf : Differentiable 𝕜 f hg : Differentiable 𝕜 g hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x x✝¹ : E hfgx : f x✝¹ = g x✝¹ A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E x : E x✝ : x ∈ univ ⊢ fderivWithin 𝕜 f univ x = fderivWithin 𝕜 g univ x

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by

simpa using hf' _

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by

Mathlib.Analysis.Calculus.MeanValue.628_0.ReDurB0qNQAwk9I

theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f f' : 𝕜 → G s : Set 𝕜 x✝ y : 𝕜 C : ℝ hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖ ≤ C hs : Convex ℝ s xs : x✝ ∈ s ys : y ∈ s x : 𝕜 hx : x ∈ s ⊢ ‖smulRight 1 (f' x)‖ ≤ ‖f' x‖

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by

simp

/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by

Mathlib.Analysis.Calculus.MeanValue.644_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f f' : 𝕜 → G s : Set 𝕜 x✝ y : 𝕜 C : ℝ≥0 hs : Convex ℝ s hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C x : 𝕜 hx : x ∈ s ⊢ ‖smulRight 1 (f' x)‖₊ ≤ ‖f' x‖₊

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by

simp

/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by

Mathlib.Analysis.Calculus.MeanValue.653_0.ReDurB0qNQAwk9I

/-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f f' : 𝕜 → G s : Set 𝕜 x✝ y✝ : 𝕜 hf : Differentiable 𝕜 f hf' : ∀ (x : 𝕜), deriv f x = 0 x y z : 𝕜 ⊢ fderiv 𝕜 f z = 0

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by

ext

/-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by

Mathlib.Analysis.Calculus.MeanValue.707_0.ReDurB0qNQAwk9I

/-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

case h E : Type u_1 inst✝⁷ : NormedAddCommGroup E inst✝⁶ : NormedSpace ℝ E F : Type u_2 inst✝⁵ : NormedAddCommGroup F inst✝⁴ : NormedSpace ℝ F 𝕜 : Type u_3 G : Type u_4 inst✝³ : IsROrC 𝕜 inst✝² : NormedSpace 𝕜 E inst✝¹ : NormedAddCommGroup G inst✝ : NormedSpace 𝕜 G f f' : 𝕜 → G s : Set 𝕜 x✝ y✝ : 𝕜 hf : Differentiable 𝕜 f hf' : ∀ (x : 𝕜), deriv f x = 0 x y z : 𝕜 ⊢ (fderiv 𝕜 f z) 1 = 0 1

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext;

simp [← deriv_fderiv, hf']

/-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext;

Mathlib.Analysis.Calculus.MeanValue.707_0.ReDurB0qNQAwk9I

/-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by

let h x := (g b - g a) * f x - (f b - f a) * g x

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x

have hI : h a = h b := by simp only; ring

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ h a = h b

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by

simp only

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x ⊢ (g b - g a) * f a - (f b - f a) * g a = (g b - g a) * f b - (f b - f a) * g b

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only;

ring

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only;

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hI : h a = h b ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring

let h' x := (g b - g a) * f' x - (f b - f a) * g' x

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hI : h a = h b h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x

have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hI : h a = h b h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))

have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hI : h a = h b h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x hhc : ContinuousOn h (Icc a b) ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)

rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

case intro.intro E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) h : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x hI : h a = h b h' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x hhc : ContinuousOn h (Icc a b) c : ℝ cmem : c ∈ Ioo a b hc : h' c = 0 ⊢ ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩

exact ⟨c, cmem, sub_eq_zero.1 hc⟩

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩

Mathlib.Analysis.Calculus.MeanValue.728_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) ⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by

let h x := (lgb - lga) * f x - (lfb - lfa) * g x

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x ⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x

have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x ⊢ Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by

have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga)) ⊢ Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga))

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)

convert this using 2

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

case h.e'_5.h.e'_3 E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x this : Tendsto h (𝓝[>] a) (𝓝 ((lgb - lga) * lfa - (lfb - lfa) * lga)) ⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2

ring

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga)) ⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring

have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb) convert this using 2 ring

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga)) ⊢ Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by

have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga)) this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb)) ⊢ Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga))

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)

convert this using 2

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

case h.e'_5.h.e'_3 E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga)) this : Tendsto h (𝓝[<] b) (𝓝 ((lgb - lga) * lfb - (lfb - lfa) * lgb)) ⊢ lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb) convert this using 2

ring

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb) convert this using 2

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue

E : Type u_1 inst✝³ : NormedAddCommGroup E inst✝² : NormedSpace ℝ E F : Type u_2 inst✝¹ : NormedAddCommGroup F inst✝ : NormedSpace ℝ F f f' : ℝ → ℝ a b : ℝ hab : a < b hfc : ContinuousOn f (Icc a b) hff'✝ : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hfd : DifferentiableOn ℝ f (Ioo a b) g g' : ℝ → ℝ hgc : ContinuousOn g (Icc a b) hgg'✝ : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hgd : DifferentiableOn ℝ g (Ioo a b) lfa lga lfb lgb : ℝ hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x hfa : Tendsto f (𝓝[>] a) (𝓝 lfa) hga : Tendsto g (𝓝[>] a) (𝓝 lga) hfb : Tendsto f (𝓝[<] b) (𝓝 lfb) hgb : Tendsto g (𝓝[<] b) (𝓝 lgb) h : ℝ → ℝ := fun x => (lgb - lga) * f x - (lfb - lfa) * g x hha : Tendsto h (𝓝[>] a) (𝓝 (lgb * lfa - lfb * lga)) hhb : Tendsto h (𝓝[<] b) (𝓝 (lgb * lfa - lfb * lga)) ⊢ ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

/- Copyright (c) 2019 Sébastien Gouëzel. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Sébastien Gouëzel, Yury Kudryashov -/ import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Slope import Mathlib.Analysis.Convex.Normed import Mathlib.Data.IsROrC.Basic import Mathlib.Topology.Instances.RealVectorSpace #align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe" /-! # The mean value inequality and equalities In this file we prove the following facts: * `Convex.norm_image_sub_le_of_norm_deriv_le` : if `f` is differentiable on a convex set `s` and the norm of its derivative is bounded by `C`, then `f` is Lipschitz continuous on `s` with constant `C`; also a variant in which what is bounded by `C` is the norm of the difference of the derivative from a fixed linear map. This lemma and its versions are formulated using `IsROrC`, so they work both for real and complex derivatives. * `image_le_of*`, `image_norm_le_of_*` : several similar lemmas deducing `f x ≤ B x` or `‖f x‖ ≤ B x` from upper estimates on `f'` or `‖f'‖`, respectively. These lemmas differ by their assumptions: * `of_liminf_*` lemmas assume that limit inferior of some ratio is less than `B' x`; * `of_deriv_right_*`, `of_norm_deriv_right_*` lemmas assume that the right derivative or its norm is less than `B' x`; * `of_*_lt_*` lemmas assume a strict inequality whenever `f x = B x` or `‖f x‖ = B x`; * `of_*_le_*` lemmas assume a non-strict inequality everywhere on `[a, b)`; * name of a lemma ends with `'` if (1) it assumes that `B` is continuous on `[a, b]` and has a right derivative at every point of `[a, b)`, and (2) the lemma has a counterpart assuming that `B` is differentiable everywhere on `ℝ` * `norm_image_sub_le_*_segment` : if derivative of `f` on `[a, b]` is bounded above by a constant `C`, then `‖f x - f a‖ ≤ C * ‖x - a‖`; several versions deal with right derivative and derivative within `[a, b]` (`HasDerivWithinAt` or `derivWithin`). * `Convex.is_const_of_fderivWithin_eq_zero` : if a function has derivative `0` on a convex set `s`, then it is a constant on `s`. * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. * `exists_hasDerivAt_eq_slope` and `exists_deriv_eq_slope` : Lagrange's Mean Value Theorem. * `domain_mvt` : Lagrange's Mean Value Theorem, applied to a segment in a convex domain. * `Convex.image_sub_lt_mul_sub_of_deriv_lt`, `Convex.mul_sub_lt_image_sub_of_lt_deriv`, `Convex.image_sub_le_mul_sub_of_deriv_le`, `Convex.mul_sub_le_image_sub_of_le_deriv`, if `∀ x, C (</≤/>/≥) (f' x)`, then `C * (y - x) (</≤/>/≥) (f y - f x)` whenever `x < y`. * `Convex.monotoneOn_of_deriv_nonneg`, `Convex.antitoneOn_of_deriv_nonpos`, `Convex.strictMono_of_deriv_pos`, `Convex.strictAnti_of_deriv_neg` : if the derivative of a function is non-negative/non-positive/positive/negative, then the function is monotone/antitone/strictly monotone/strictly monotonically decreasing. * `convexOn_of_deriv`, `convexOn_of_deriv2_nonneg` : if the derivative of a function is increasing or its second derivative is nonnegative, then the original function is convex. * `hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt` : a C^1 function over the reals is strictly differentiable. (This is a corollary of the mean value inequality.) -/ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] open Metric Set Asymptotics ContinuousLinearMap Filter open scoped Classical Topology NNReal /-! ### One-dimensional fencing inequalities -/ /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by change Icc a b ⊆ { x | f x ≤ B x } set s := { x | f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_forall_exists_gt ha rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy cases' hxB.lt_or_eq with hxB hxB · -- If `f x < B x`, then all we need is continuity of both sides refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x := A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this exact this.mono fun y => le_of_lt · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ specialize hf' x xab r hfr have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z := (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB) obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists refine' ⟨z, _, hz⟩ have := (hfz.trans hzB).le rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this #align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)` is bounded above by `B'`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) -- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x` (bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound · rwa [sub_self, mul_zero, add_zero] · exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) · intro x hx exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) · intro x _ _ rw [mul_one] exact (lt_add_iff_pos_right _).2 hr exact hx intro x hx have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 := continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const) convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp #align image_le_of_liminf_slope_right_le_deriv_boundary image_le_of_liminf_slope_right_le_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has right derivative `B'` at every point of `[a, b)`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_le hr) ha hB hB' bound #align image_le_of_deriv_right_lt_deriv_boundary' image_le_of_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x < B' x` whenever `f x = B x`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_le_of_deriv_right_lt_deriv_boundary image_le_of_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `f a ≤ B a`; * `B` has derivative `B'` everywhere on `ℝ`; * `f` has right derivative `f'` at every point of `[a, b)`; * we have `f' x ≤ B' x` on `[a, b)`. Then `f x ≤ B x` everywhere on `[a, b]`. -/ theorem image_le_of_deriv_right_le_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, f' x ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary hf ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_le (lt_of_le_of_lt (bound x hx) hr) #align image_le_of_deriv_right_le_deriv_boundary image_le_of_deriv_right_le_deriv_boundary /-! ### Vector-valued functions `f : ℝ → E` -/ section variable {f : ℝ → E} {a b : ℝ} /-- General fencing theorem for continuous functions with an estimate on the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `B` has right derivative at every point of `[a, b)`; * for each `x ∈ [a, b)` the right-side limit inferior of `(‖f z‖ - ‖f x‖) / (z - x)` is bounded above by a function `f'`; * we have `f' x < B' x` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. -/ theorem image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary {E : Type*} [NormedAddCommGroup E] {f : ℝ → E} {f' : ℝ → ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (‖f z‖ - ‖f x‖) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope (norm ∘ f) x z < r) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_lt_deriv_boundary' (continuous_norm.comp_continuousOn hf) hf' ha hB hB' bound #align image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_liminf_right_slope_norm_lt_deriv_boundary hf (fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le hr) ha hB hB' bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary' image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * the norm of `f'` is strictly less than `B'` whenever `‖f x‖ = B x`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_lt_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f x‖ = B x → ‖f' x‖ < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_lt_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_lt_deriv_boundary image_norm_le_of_norm_deriv_right_lt_deriv_boundary /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` and `B` have right derivatives `f'` and `B'` respectively at every point of `[a, b)`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary' {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ContinuousOn B (Icc a b)) (hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_le_of_liminf_slope_right_le_deriv_boundary (continuous_norm.comp_continuousOn hf) ha hB hB' fun x hx _ hr => (hf' x hx).liminf_right_slope_norm_le ((bound x hx).trans_lt hr) #align image_norm_le_of_norm_deriv_right_le_deriv_boundary' image_norm_le_of_norm_deriv_right_le_deriv_boundary' /-- General fencing theorem for continuous functions with an estimate on the norm of the derivative. Let `f` and `B` be continuous functions on `[a, b]` such that * `‖f a‖ ≤ B a`; * `f` has right derivative `f'` at every point of `[a, b)`; * `B` has derivative `B'` everywhere on `ℝ`; * we have `‖f' x‖ ≤ B x` everywhere on `[a, b)`. Then `‖f x‖ ≤ B x` everywhere on `[a, b]`. We use one-sided derivatives in the assumptions to make this theorem work for piecewise differentiable functions. -/ theorem image_norm_le_of_norm_deriv_right_le_deriv_boundary {f' : ℝ → E} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) {B B' : ℝ → ℝ} (ha : ‖f a‖ ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ B' x) : ∀ ⦃x⦄, x ∈ Icc a b → ‖f x‖ ≤ B x := image_norm_le_of_norm_deriv_right_le_deriv_boundary' hf hf' ha (fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound #align image_norm_le_of_norm_deriv_right_le_deriv_boundary image_norm_le_of_norm_deriv_right_le_deriv_boundary /-- A function on `[a, b]` with the norm of the right derivative bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`. -/ theorem norm_image_sub_le_of_norm_deriv_right_le_segment {f' : ℝ → E} {C : ℝ} (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by let g x := f x - f a have hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const have hg' : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x := by intro x hx simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _) let B x := C * (x - a) have hB : ∀ x, HasDerivAt B C x := by intro x simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a)) convert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound simp only; rw [sub_self, norm_zero, sub_self, mul_zero] #align norm_image_sub_le_of_norm_deriv_right_le_segment norm_image_sub_le_of_norm_deriv_right_le_segment /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound exact (hf x <| Ico_subset_Icc_self hx).mono_of_mem (Icc_mem_nhdsWithin_Ici hx) #align norm_image_sub_le_of_norm_deriv_le_segment' norm_image_sub_le_of_norm_deriv_le_segment' /-- A function on `[a, b]` with the norm of the derivative within `[a, b]` bounded by `C` satisfies `‖f x - f a‖ ≤ C * (x - a)`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment {C : ℝ} (hf : DifferentiableOn ℝ f (Icc a b)) (bound : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ C) : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a) := by refine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound exact fun x hx => (hf x hx).hasDerivWithinAt #align norm_image_sub_le_of_norm_deriv_le_segment norm_image_sub_le_of_norm_deriv_le_segment /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `HasDerivWithinAt` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01' {f' : ℝ → E} {C : ℝ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivWithinAt f (f' x) (Icc (0 : ℝ) 1) x) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖f' x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01' norm_image_sub_le_of_norm_deriv_le_segment_01' /-- A function on `[0, 1]` with the norm of the derivative within `[0, 1]` bounded by `C` satisfies `‖f 1 - f 0‖ ≤ C`, `derivWithin` version. -/ theorem norm_image_sub_le_of_norm_deriv_le_segment_01 {C : ℝ} (hf : DifferentiableOn ℝ f (Icc (0 : ℝ) 1)) (bound : ∀ x ∈ Ico (0 : ℝ) 1, ‖derivWithin f (Icc (0 : ℝ) 1) x‖ ≤ C) : ‖f 1 - f 0‖ ≤ C := by simpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one) #align norm_image_sub_le_of_norm_deriv_le_segment_01 norm_image_sub_le_of_norm_deriv_le_segment_01 theorem constant_of_has_deriv_right_zero (hcont : ContinuousOn f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt f 0 (Ici x) x) : ∀ x ∈ Icc a b, f x = f a := by have : ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ 0 * (x - a) := fun x hx => norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this #align constant_of_has_deriv_right_zero constant_of_has_deriv_right_zero theorem constant_of_derivWithin_zero (hdiff : DifferentiableOn ℝ f (Icc a b)) (hderiv : ∀ x ∈ Ico a b, derivWithin f (Icc a b) x = 0) : ∀ x ∈ Icc a b, f x = f a := by have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx => norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx #align constant_of_deriv_within_zero constant_of_derivWithin_zero variable {f' g : ℝ → E} /-- If two continuous functions on `[a, b]` have the same right derivative and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_has_deriv_right_eq (derivf : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x) (derivg : ∀ x ∈ Ico a b, HasDerivWithinAt g (f' x) (Ici x) x) (fcont : ContinuousOn f (Icc a b)) (gcont : ContinuousOn g (Icc a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by simp only [← @sub_eq_zero _ _ (f _)] at hi ⊢ exact hi ▸ constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by simpa only [sub_self] using (derivf y hy).sub (derivg y hy) #align eq_of_has_deriv_right_eq eq_of_has_deriv_right_eq /-- If two differentiable functions on `[a, b]` have the same derivative within `[a, b]` everywhere on `[a, b)` and are equal at `a`, then they are equal everywhere on `[a, b]`. -/ theorem eq_of_derivWithin_eq (fdiff : DifferentiableOn ℝ f (Icc a b)) (gdiff : DifferentiableOn ℝ g (Icc a b)) (hderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)) (hi : f a = g a) : ∀ y ∈ Icc a b, f y = g y := by have A : ∀ y ∈ Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy => (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) have B : ∀ y ∈ Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy => (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.mono_of_mem (Icc_mem_nhdsWithin_Ici hy) exact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm ▸ B y hy) fdiff.continuousOn gdiff.continuousOn hi #align eq_of_deriv_within_eq eq_of_derivWithin_eq end /-! ### Vector-valued functions `f : E → G` Theorems in this section work both for real and complex differentiable functions. We use assumptions `[IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/ section variable {𝕜 G : Type*} [IsROrC 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] namespace Convex variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`. We just have to check the differentiability of the composition and bounds on its derivative, which is straightforward but tedious for lack of automation. -/ set g := (AffineMap.lineMap x y : ℝ → E) have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys have hD : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt (f ∘ g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by simpa using ((hf (g t) (segm ht)).restrictScalars ℝ).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm have bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖f' (g t) (y - x)‖ ≤ C * ‖y - x‖ := fun t ht => le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _ simpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasFDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasFDerivWithin_le {C : ℝ≥0} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := by rw [lipschitzOnWith_iff_norm_sub_le] intro x x_in y y_in exact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in #align convex.lipschitz_on_with_of_nnnorm_has_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is `K`-Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt` for a version that claims existence of `K` instead of an explicit estimate. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) (K : ℝ≥0) (hK : ‖f' x‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by obtain ⟨ε, ε0, hε⟩ : ∃ ε > 0, ball x ε ∩ s ⊆ { y | HasFDerivWithinAt f (f' y) s y ∧ ‖f' y‖₊ < K } exact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK)) rw [inter_comm] at hε refine' ⟨s ∩ ball x ε, inter_mem_nhdsWithin _ (ball_mem_nhds _ ε0), _⟩ exact (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun y hy => (hε hy).1.mono (inter_subset_left _ _)) fun y hy => (hε hy).2.le #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at_of_nnnorm_lt Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt /-- Let `s` be a convex set in a real normed vector space `E`, let `f : E → G` be a function differentiable within `s` in a neighborhood of `x : E` with derivative `f'`. Suppose that `f'` is continuous within `s` at `x`. Then for any number `K : ℝ≥0` larger than `‖f' x‖₊`, `f` is Lipschitz on some neighborhood of `x` within `s`. See also `Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt` for a version with an explicit estimate on the Lipschitz constant. -/ theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) : ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := (exists_gt _).imp <| hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt hder hcont #align convex.exists_nhds_within_lipschitz_on_with_of_has_fderiv_within_at Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt /-- The mean value theorem on a convex set: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le Convex.norm_image_sub_le_of_norm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_within_le Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le Convex.norm_image_sub_le_of_norm_fderiv_le /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `fderiv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_fderiv_le Convex.lipschitzOnWith_of_nnnorm_fderiv_le /-- The mean value theorem: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `fderiv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E rw [← lipschitzOn_univ] exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ /-- Variant of the mean value inequality on a convex set, using a bound on the difference between the derivative and a fixed linear map, rather than a bound on the derivative itself. Version with `HasFDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasFDerivWithin_le' (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := by /- We subtract `φ` to define a new function `g` for which `g' = 0`, for which the previous theorem applies, `Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le`. Then, we just need to glue together the pieces, expressing back `f` in terms of `g`. -/ let g y := f y - φ y have hg : ∀ x ∈ s, HasFDerivWithinAt g (f' x - φ) s x := fun x xs => (hf x xs).sub φ.hasFDerivWithinAt calc ‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp _ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel _ = ‖g y - g x‖ := by simp _ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys #align convex.norm_image_sub_le_of_norm_has_fderiv_within_le' Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderivWithin`. -/ theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_within_le' Convex.norm_image_sub_le_of_norm_fderivWithin_le' /-- Variant of the mean value inequality on a convex set. Version with `fderiv`. -/ theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasFDerivWithin_le' (fun x hx => (hf x hx).hasFDerivAt.hasFDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_fderiv_le' Convex.norm_image_sub_le_of_norm_fderiv_le' /-- If a function has zero Fréchet derivative at every point of a convex set, then it is a constant on this set. -/ theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y := by have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl] simpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy #align convex.is_const_of_fderiv_within_eq_zero Convex.is_const_of_fderivWithin_eq_zero theorem _root_.is_const_of_fderiv_eq_zero {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0) (x y : E) : f x = f y := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial #align is_const_of_fderiv_eq_zero is_const_of_fderiv_eq_zero /-- If two functions have equal Fréchet derivatives at every point of a convex set, and are equal at one point in that set, then they are equal on that set. -/ theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) : s.EqOn f g := fun y hy => by suffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this refine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy rw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz] #align convex.eq_on_of_fderiv_within_eq Convex.eqOn_of_fderivWithin_eq theorem _root_.eq_of_fderiv_eq {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx #align eq_of_fderiv_eq eq_of_fderiv_eq end Convex namespace Convex variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/ theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs xs ys #align convex.norm_image_sub_le_of_norm_has_deriv_within_le Convex.norm_image_sub_le_of_norm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `HasDerivWithinAt` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) : LipschitzOnWith C f s := Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) (fun x hx => le_trans (by simp) (bound x hx)) hs #align convex.lipschitz_on_with_of_nnnorm_has_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function within this set is bounded by `C`, then the function is `C`-Lipschitz. Version with `derivWithin` -/ theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_within_le Convex.norm_image_sub_le_of_norm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `derivWithin` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_within_le Convex.lipschitzOnWith_of_nnnorm_derivWithin_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv`. -/ theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound xs ys #align convex.norm_image_sub_le_of_norm_deriv_le Convex.norm_image_sub_le_of_norm_deriv_le /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is bounded by `C` on `s`, then the function is `C`-Lipschitz on `s`. Version with `deriv` and `LipschitzOnWith`. -/ theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s := hs.lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivAt.hasDerivWithinAt) bound #align convex.lipschitz_on_with_of_nnnorm_deriv_le Convex.lipschitzOnWith_of_nnnorm_deriv_le /-- The mean value theorem set in dimension 1: if the derivative of a function is bounded by `C`, then the function is `C`-Lipschitz. Version with `deriv` and `LipschitzWith`. -/ theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f := lipschitzOn_univ.1 <| convex_univ.lipschitzOnWith_of_nnnorm_deriv_le (fun x _ => hf x) fun x _ => bound x #align lipschitz_with_of_nnnorm_deriv_le lipschitzWith_of_nnnorm_deriv_le /-- If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero, then it is a constant function. -/ theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0) (x y : 𝕜) : f x = f y := is_const_of_fderiv_eq_zero hf (fun z => by ext; simp [← deriv_fderiv, hf']) _ _ #align is_const_of_deriv_eq_zero is_const_of_deriv_eq_zero end Convex end /-! ### Functions `[a, b] → ℝ`. -/ section Interval -- Declare all variables here to make sure they come in a correct order variable (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hfd : DifferentiableOn ℝ f (Ioo a b)) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Icc a b)) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hgd : DifferentiableOn ℝ g (Ioo a b)) /-- Cauchy's **Mean Value Theorem**, `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope : ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by let h x := (g b - g a) * f x - (f b - f a) * g x have hI : h a = h b := by simp only; ring let h' x := (g b - g a) * f' x - (f b - f a) * g' x have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx => ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)) have hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc) rcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with ⟨c, cmem, hc⟩ exact ⟨c, cmem, sub_eq_zero.1 hc⟩ #align exists_ratio_has_deriv_at_eq_ratio_slope exists_ratio_hasDerivAt_eq_ratio_slope /-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb) convert this using 2 ring

let h' x := (lgb - lga) * f' x - (lfb - lfa) * g' x

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c := by let h x := (lgb - lga) * f x - (lfb - lfa) * g x have hha : Tendsto h (𝓝[>] a) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[>] a) (𝓝 <| (lgb - lga) * lfa - (lfb - lfa) * lga) := (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga) convert this using 2 ring have hhb : Tendsto h (𝓝[<] b) (𝓝 <| lgb * lfa - lfb * lga) := by have : Tendsto h (𝓝[<] b) (𝓝 <| (lgb - lga) * lfb - (lfb - lfa) * lgb) := (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb) convert this using 2 ring

Mathlib.Analysis.Calculus.MeanValue.742_0.ReDurB0qNQAwk9I

/-- Cauchy's **Mean Value Theorem**, extended `HasDerivAt` version. -/ theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) : ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c

Mathlib_Analysis_Calculus_MeanValue