Lipschitz function

2020 Mathematics Subject Classification: Primary: 54E40 [MSN][ZBL]

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 $f\in \operatorname{Lip} ([a,b])$. The least constant in $\eqref{eq:1}$ is called Lipschitz constant.

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 $$\begin{equation}\label{eq:2} \delta (f(x), f(y)) \leq M d (x,y) \qquad\qquad \forall x,y\in X\, . \end{equation}$$ The Lipschitz constant of $f$, usually denoted by ${\rm Lip}\, (f)$ is the least constant $M$ for which the inequality $\eqref{eq:2}$ is valid.

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 $U\subset\mathbb R^n$ is an open set), then $f$ is differentiable almost everywhere (Rademacher theorem). Another important theorem about Lipschitz functions between euclidean spaces is Kirszbraun's extension theorem. See Lipschitz condition for more details.