Theorems · Theorem · real analysis

Convex.norm_image_sub_le_of_norm_deriv_le

∀ {𝕜 : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup G] [inst_2 : NormedSpace 𝕜 G] {f : 𝕜 → G}
  {s : Set 𝕜} {x y : 𝕜} {C : ℝ},
  (∀ x ∈ s, DifferentiableAt 𝕜 f x) →
    (∀ x ∈ s, ‖deriv f x‖ ≤ C) → Convex ℝ s → x ∈ s → y ∈ s → ‖f y - f x‖ ≤ C * ‖y - x‖

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.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

2 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpace