Theorems · Theorem · real analysis

Convex.lipschitzOnWith_of_nnnorm_hasFDerivWithin_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} {f' : E → E →L[𝕜] G} {C : NNReal},
  (∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) → (∀ x ∈ s, ‖f' 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 HasFDerivWithinAt and LipschitzOnWith.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

5 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNontriviallyNormedFieldIsRCLikeNormedFieldNormedSpaceNormedAddCommGroupNormedSpace