Difference between revisions of "Lipschitz constant"
From Encyclopedia of Mathematics
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\begin{equation*} | \begin{equation*} | ||
| − | |f(y)-f(x)|\leq M|y-x|^{\alpha}, \quad x,y\in[a,b] | + | |f(y)-f(x)|\leq M|y-x|^{\alpha}, \quad x,y\in[a,b]. |
| + | \end{equation*} | ||
| + | |||
| + | |||
| + | In other words, one can find | ||
| + | \begin{equation*} | ||
| + | M = \sup\limits_{x,y\in[a,b]}\frac{|f(y)-f(x)|}{|y-x|^{\alpha}}. | ||
\end{equation*} | \end{equation*} | ||
If $\alpha=1$ $f$ is called [[Lipschitz Function | Lipschitz function]]. | If $\alpha=1$ $f$ is called [[Lipschitz Function | Lipschitz function]]. | ||
Revision as of 16:14, 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*}
In other words, one can find
\begin{equation*}
M = \sup\limits_{x,y\in[a,b]}\frac{|f(y)-f(x)|}{|y-x|^{\alpha}}.
\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=28853
Lipschitz constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lipschitz_constant&oldid=28853
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article