Difference between revisions of "Lipschitz function"
(Added def of Lip constant, Kirszbraun's theorem, Category and MSC classification) |
m |
||
| Line 7: | Line 7: | ||
|f(x)-f(y)| \leq M|x-y|. | |f(x)-f(y)| \leq M|x-y|. | ||
\end{equation} | \end{equation} | ||
| − | Then the function $f$ is called Lipschitz on $[a,b]$, and one writes $f\in \operatorname{Lip} | + | 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 space|metric spaces]] $(X,d)$ and $(Y, \delta)$: such maps | The concept can be readily extended to general maps $f$ between two [[Metric space|metric spaces]] $(X,d)$ and $(Y, \delta)$: such maps | ||
| Line 22: | Line 22: | ||
===Properties=== | ===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 theorem|Kirszbraun's extension theorem]]. See [[Lipschitz condition]] for more details. | |
| − | 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 theorem|Kirszbraun's extension theorem]]. | ||
Latest revision as of 16:49, 9 November 2013
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.
Lipschitz function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_function&oldid=30450