---
tag: 0TMZ21WKCN
kind: definition
name: Lipschitz map
aliases: [lipschitz-function]
msc: [26A16]
defines: 0AW1E6YTJY
uses: [063W5JMY85]
sources:
  - url: https://en.wikipedia.org/wiki/Lipschitz_continuity
    locator: Definitions
---

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], 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|0MT2S0ZX4Y]].
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 $K$.