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Difference between revisions of "Lipschitz constant"

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''for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597001.png" /> defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597002.png" />''
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''for a function $f$ defined on an interval $[a,b]$''
  
The greatest lower bound of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597003.png" /> in the [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597005.png" />,
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The greatest lower bound of constants $M>0$ in the [[Lipschitz condition|Lipschitz condition]] of order $\alpha$, $0<\alpha\leq1$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059700/l0597006.png" /></td> </tr></table>
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\begin{equation*}
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|f(y)-f(x)|\leq M|y-x|^{\alpha}, \quad x,y\in[a,b]
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\end{equation*}
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If $\alpha=1$ $f$ is called [[Lipschitz Function | Lipschitz function]].

Revision as of 16:09, 23 November 2012

for a function $f$ defined on an interval $[a,b]$

The greatest lower bound of constants $M>0$ in the Lipschitz condition of order $\alpha$, $0<\alpha\leq1$,

\begin{equation*} |f(y)-f(x)|\leq M|y-x|^{\alpha}, \quad x,y\in[a,b] \end{equation*}

If $\alpha=1$ $f$ is called Lipschitz function.

How to Cite This Entry:
Lipschitz constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_constant&oldid=13226
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article