Answer to Question #211099 in Real Analysis for Digger

Question

Let k ≥ 0 and f : M → M a k-Lipschitz function. Let ε > 0. Give the largest

number φ > 0, if any, such that ∀x, y ∈ M, d(x, y) < φ implies d(f(x), d(y)) < ε.

Answer 1

Let f : A → R, A ⊂ Rⁿ, be an L-Lipschitz function. Then there exists an L-Lipschitz function F : Rⁿ → R such that F|A = f.

Proof. Because the functions

$f_a (x) := f(a) + L|x - a| , a ∈ A$ ,

are L-Lipschitz on Rn, the function

$F(x) :=\infin_{ a∈A} f_a (x) , F : R^n → R ,$

is L-Lipschitz by Lemma 2.1. It is obvious that F(a) = f(a) whenever a ∈ A.

The extension F in Theorem 2.3 is the largest L-Lipschitz extension of f in the sense that if G : Rn → R is L-Lipschitz and G|A = f, then G ≤ F. One can also find the smallest L-Lipschitz extension of f, by setting

$F(x) := sup_{ a∈A} f(a) - L|x - a| , x ∈ R$