Theorems · Theorem · global analysis

HasLineDerivAt.le_of_lipschitzOn

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

Converse to the mean value inequality: if f is line differentiable at x₀ and C-lipschitz on a neighborhood of x₀ then its line derivative at x₀ in the direction v has norm bounded by C * ‖v‖. This version only assumes that ‖f x - f x₀‖ ≤ C * ‖x - x₀‖ in a neighborhood of x.

Defined in

Mathlib.Analysis.Calculus.LineDeriv.Basic

Cited by

1 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace