Lipschitz continuity

A quantitative continuity condition: |f(x)-f(y)| ≤ L|x-y| for some constant L.

Lipschitz continuity

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:E\to Y$ with $E\subseteq X$ . The function $f$ is Lipschitz continuous on $E$ if there exists a constant $L\in[0,\infty)$ such that

\forall x,y\in E,\ d_Y\!\bigl(f(x),f(y)\bigr)\le L\, d_X(x,y).

Any such $L$ is called a Lipschitz constant for $f$ on $E$ .

Lipschitz continuity is stronger than uniform continuity and provides explicit control of error propagation. It is ubiquitous in analysis and differential equations.

Examples:

$f:\mathbb{R}\to\mathbb{R}$ , $f(x)=ax+b$ is Lipschitz with constant $L=|a|$ .
On $\mathbb{R}$ , the function $f(x)=|x|$ is Lipschitz with constant $L=1$ .
$f(x)=\sqrt{x}$ is not Lipschitz on $[0,1]$ (the slope blows up near $0$ ), but it is Lipschitz on $[\delta,1]$ for any fixed $\delta>0$ .