Theorems · Theorem · global analysis

HasFDerivAt.le_of_lipschitzOn

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {f' : E →L[𝕜] F} {x₀ : E},
  HasFDerivAt f f' x₀ → ∀ {s : Set E}, s ∈ nhds x₀ → ∀ {C : NNReal}, LipschitzOnWith C f s → ‖f'‖ ≤ ↑C

Converse to the mean value inequality: if f is differentiable at x₀ and C-lipschitz on a neighborhood of x₀ then its derivative at x₀ has norm bounded by C.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Basic

Cited by

2 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace