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Lipschitz function (Definition)

Let $ W \subseteq X \subseteq \mathbb{C}$ and $ f\colon X\to\mathbb{C}$. Then $ f$ is Lipschitz on $ W$ if there exists an $ M\in\mathbb{R}$ such that, for all $ x,y\in W$, $ x \neq y$

$\displaystyle \vert f(x)-f(y)\vert\leq M\vert x-y\vert$

If $ a,b\in\mathbb{R}$ with $ a<b$ and $ f\colon [a,b]\to\mathbb{R}$ is Lipschitz on $ (a,b)$, then $ f$ is absolutely continuous on $ [a,b]$.

Example: Is

$\displaystyle f(x) = \frac{1}{\sqrt{x}},~~~x \in [0,1]$

a Lipschitz function.

We need to estimate the constant $ M$.

$\displaystyle \vert f(x) - f(y)\vert = \left\vert\frac{1}{\sqrt{x}} - \frac{1}{... ...ight\vert = \frac{1}{\vert\sqrt{xy} (\sqrt{x} + \sqrt{y})\vert} \vert x-y\vert.$

It follows that

$\displaystyle M = \frac{1}{\vert\sqrt{xy} (\sqrt{x} + \sqrt{y})\vert} $

and $ f(x)$ is not Lipschitz at $ x=0$.



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"Lipschitz function" is owned by bwebste. [ full author list (5) | owner history (1) ]
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Cross-references: estimate, absolutely continuous
There are 6 references to this entry.

This is version 9 of Lipschitz function, born on 2003-10-15, modified 2008-02-19.
Object id is 5054, canonical name is LipschitzFunction.
Accessed 6046 times total.

Classification:
AMS MSC: 26A16 (Real functions :: Functions of one variable :: Lipschitz classes)
Pending Errata and Addenda
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Discussion
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about Lipschitz function by perucho on 2004-07-03 02:21:02
Lipschitz function is a particular case of functions uniformly continous where the module of continuity has the form:
$\delta(\epsilon)=c\epsilon$.
It is a special case(\mu=1) of Holder(o with dieresis) function:
$\vertf(x)-f(y)\vert\leqM\vertx-y\vert^\mu$,
with $0<\mu\leq1$.
A classic example of no-lipschitzian but holderian fuction is
$f(x)=\sqrt{x},x\geq0$ where we can put $M=1,\mu=1/2$, with $y=0$.
Lipschitz function can be generalized to several variables. Let
$f:E\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$.
We say that $f$ is lipschitzian locally in $E$ if for every point $x\inE$ there exists an open ball $B(x,r)$(where $r$ can depend on $x$) and a constant $M$(that also it can depend on $x$) such that, if $z,y\inB(x,r)\capE$, we have:
$\vertf(z)-f(y)\vert \leq M\vertz-y\vert$.
$f$ will be lipschitzian globally if the latter condition is valid $\forallz,y\inE$.
Lipchitz function has importance in some theorems, for instance in
Picard-Lindel$\ddot{o}$f theorem.(Such Lipschitz's condition can be eliminated in a theorem more general than the previous one, usually
so-called Cauchy-Peano theorem.
Pedro



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