norm_lineDeriv_le_of_lipschitz — Mathlib

English

If a bound holds for all x near x0, then the line derivative satisfies the same bound at x0 for all directions v.

Русский

Если.bound верно для всех x около x0, то линейная производная удовлетворяет такое же ограничение в x0 по любому направлению v.

LaTeX

$$If the bound holds for all x near x0, then for every v, |lineDeriv(f,x0,v)| ≤ bound$$

Lean4

/-- Converse to the mean value inequality: if `f` is `C`-lipschitz then
its line derivative at `x₀` in the direction `v` has norm bounded by `C * ‖v‖`.
Version using `lineDeriv`. -/
theorem norm_lineDeriv_le_of_lipschitz {f : E → F} {x₀ : E} {C : ℝ≥0} (hlip : LipschitzWith C f) :
    ‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖ :=
  norm_lineDeriv_le_of_lipschitzOn 𝕜 univ_mem (lipschitzOnWith_univ.2 hlip)

Похожие утверждения