Lipschitz function

Let a function $f:[a,b]\to \mathbb R$ be such that for some constant M and for all $x,y\in [a,b]$ \begin{equation}\label{eq:1} |f(x)-f(y)| \leq M|x-y|. \end{equation} Then the function $f$ is called Lipschitz on $[a,b]$, and one writes $f\in \operatorname{Lip}_M[a,b]$.

The concept can be readily extended to general maps $f$ between two metric spaces $(X,d)$ and $(Y, \delta)$: such maps are called Lipschitz if for some constant $M$ one has \begin{equation}\label{eq:2} \delta (f(x), f(y)) \leq M d (x,y) \qquad\qquad \forall x,y\in X\, . \end{equation}

A mapping $f:X\to Y$ is called bi-Lipschitz if it is Lipschitz and has an inverse mapping $f^{-1}:f(X)\to X$ which is also Lipschitz.

Lipschitz maps play a fundamental role in several areas of mathematics like, for instance, Partial differential equations, Metric geometry and Geometric measure theory.

Properties

If a mapping $f:U\to \mathbb R^k$ is Lipschitz (where open set $U\subset\mathbb R^n$), then $f$ is differentiable almost everywhere (Rademacher theorem).