Theorems · Theorem · real analysis

Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4}
  [inst_2 : NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → G} {s : Set E} {C : NNReal},
  DifferentiableOn 𝕜 f s → (∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) → Convex ℝ s → LipschitzOnWith C f s

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.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

0 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupNormedSpaceNontriviallyNormedFieldIsRCLikeNormedFieldNormedSpaceNormedAddCommGroupNormedSpace