View source for 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 space|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 equation|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]]).