English
If f is Lipschitz on s with constant C, then ||deriv f x0|| ≤ C for x0 ∈ s neighborhood.
Русский
Если f лпшицева на mове s с константой C, то ||deriv f x0|| ≤ C при x0 ∈ s
LaTeX
$$$\\|\\operatorname{deriv}f(x_0)\\| \\le C \\quad \\text{whenever } x_0 \\in s$ and $C$-LipschitzOnWith C f s$$
Lean4
/-- Converse to the mean value inequality: if `f` is `C`-lipschitz
on a neighborhood of `x₀` then its derivative at `x₀` has norm bounded by `C`. This version
only assumes that `‖f x - f x₀‖ ≤ C * ‖x - x₀‖` in a neighborhood of `x`. -/
theorem norm_deriv_le_of_lip' {f : 𝕜 → F} {x₀ : 𝕜} {C : ℝ} (hC₀ : 0 ≤ C)
(hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖deriv f x₀‖ ≤ C := by
simpa [norm_deriv_eq_norm_fderiv] using norm_fderiv_le_of_lip' 𝕜 hC₀ hlip
