Theorems · Theorem · real analysis

Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le

∀ {𝕜 : Type u_3} {G : Type u_4} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup G] [inst_2 : NormedSpace 𝕜 G]
  {f f' : 𝕜 → G} {s : Set 𝕜} {C : NNReal},
  Convex ℝ s → (∀ x ∈ s, HasDerivWithinAt f (f' x) s x) → (∀ x ∈ s, ‖f' x‖₊ ≤ C) → LipschitzOnWith C f s

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.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

3 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupNormedSpace