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Difference between revisions of "Lipschitz function"

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{{MSC|54E40}}
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[[Category:Analysis]]
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Let a function $f:[a,b]\to \mathbb R$ be such that for some constant M and for all $x,y\in [a,b]$
 
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}
 
\begin{equation}\label{eq:1}
 
|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 $f\in \operatorname{Lip}_M[a,b]$.
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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
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\delta (f(x), f(y)) \leq M d (x,y) \qquad\qquad \forall x,y\in X\, .
 
\delta (f(x), f(y)) \leq M d (x,y) \qquad\qquad \forall x,y\in X\, .
 
\end{equation}
 
\end{equation}
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The Lipschitz constant of $f$, usually denoted by ${\rm Lip}\, (f)$ is the least constant $M$ for which
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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.
 
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.
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===Properties===
 
===Properties===
 
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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]]).
 

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.

How to Cite This Entry:
Lipschitz function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_function&oldid=29202