---
tag: 055VVSN00V
kind: definition
name: Lipschitz condition
msc: [26A16, 34A12]
defines: 0AW1E6YTJY
uses: [0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Lipschitz_continuity
    locator: Definitions
---

## Definition

Let $S$ be a subset of $\mathbb{R}^n$, let $f : S \to \mathbb{R}^m$ be a [[function|0EPAT51DXQ]], and let $K \ge 0$ be a real number.
The function $f$ satisfies a Lipschitz condition on $S$ with constant $K$, or is $K$-Lipschitz on $S$, if
$$\| f(x) - f(y) \| \le K\, \| x - y \| \quad \text{for all } x, y \in S.$$
Let $D \subseteq \mathbb{R} \times \mathbb{R}^n$ and $f : D \to \mathbb{R}^m$. The function $f$ is Lipschitz in $x$ on $D$ with constant $K$ if $\| f(t, x) - f(t, y) \| \le K \| x - y \|$ whenever $(t, x), (t, y) \in D$.

## Remarks

A $K$-Lipschitz function is uniformly continuous.
A differentiable function on a convex set with $\| f'(x) \| \le K$ at each point is $K$-Lipschitz, by the mean value inequality. For example, $\sin$ is $1$-Lipschitz on $\mathbb{R}$.
The function $x \mapsto \sqrt{|x|}$ is continuous but not Lipschitz on any interval that contains $0$. The [[non-uniqueness example|00K7ASZ1PR]] uses a function of this kind.
The [[Picard–Lindelöf theorem|0P6KRD5B7A]] needs a Lipschitz condition in $x$.

## Explanations

### L1 human


A Lipschitz function cannot change faster than a fixed rate.
The slope of each chord of its graph is at most $K$ in absolute value.