Lipschitz continuity - Imported Knowlpedia Content

A Lipschitz continuous map between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is a map $f\colon X\to Y$ for which there exists a constant $L\ge 0$ such that for all $x,y\in X$ ,

d_Y\bigl(f(x),f(y)\bigr)\le L\,d_X(x,y).

Remarks

Lipschitz continuity is a quantitative strengthening of uniform continuity: every Lipschitz map is uniformly continuous. It also gives direct control of the size of images of sets via diameter.

Examples

On $\mathbb{R}$ with the usual metric, $f(x)=ax+b$ is Lipschitz with constant $L=|a|$ .
On any metric space $(X,d)$ , the function $x\mapsto d(x,x_0)$ (distance to a fixed point $x_0$ ) is Lipschitz with constant $1$ .