Lipschitz map

Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, let $f \colon X \to Y$ be a function and let $K \ge 0$ be a real number. The map $f$ is Lipschitz with constant $K$ when $$d_Y\bigl(f(x), f(y)\bigr) \le K \, d_X(x, y) \quad \text{for all } x, y \in X.$$ The map $f$ is Lipschitz when it is Lipschitz with some constant $K \ge 0$.

Remarks

The smallest such $K$ is the Lipschitz constant of $f$. The map $x \mapsto |x|$ on $\mathbb{R}$ is Lipschitz with constant $1$. The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is not Lipschitz, because $\sqrt{x} / x$ is not bounded near $0$. A Lipschitz map with constant $K < 1$ is a contraction. A differentiable $f \colon \mathbb{R} \to \mathbb{R}$ with $|f'(x)| \le K$ for all $x$ is Lipschitz with constant $K$, by the mean value theorem.

Explanations

L1 human

A Lipschitz map cannot stretch distances by more than the factor $K$. For a real function, this means that the slopes of all its chords are between $-K$ and $K$.