Theorems · Theorem · real analysis

lipschitzWith_of_nnnorm_fderiv_le

∀ {𝕜 : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]
  [inst_2 : NormedAddCommGroup G] [inst_3 : NormedSpace 𝕜 G] {E : Type u_5} [inst_4 : NormedAddCommGroup E]
  [inst_5 : NormedSpace 𝕜 E] {f : E → G} {C : NNReal},
  Differentiable 𝕜 f → (∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C) → LipschitzWith C f

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.

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

1 results in Mathlib

Foundations

Depth 194 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldIsRCLikeNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace