Lipschitz conditionLipschitzCondition2001-11-12 13:49:132005-02-28 10:42:45DefinitionHolderHolder continuousLipschitz constant\usepackage{amsmath}
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\newtheorem{theorem}[proposition]{Theorem}A mapping $f: X \to Y$ between metric spaces is said to satisfy the
Lipschitz condition, or to be \emph{Lipschitz continuous} or \emph{$L$-Lipschitz} if there exists a real constant $L$ such
that
$$ d_Y(f(p),f(q)) \leq L d_X(p,q),\quad \text{for all}\; p,q\in X.$$
The least constant $L$ for which the previous inequality holds, is called the \emph{Lipschitz constant} of $f$.
The space of Lipschitz continuous functions is often denoted by $\mathrm{Lip}(X,Y)$.
Clearly, every Lipschitz continuous function is continuous.
\paragraph{Notes.}
More generally, one says that a mapping satisfies
a Lipschitz condition of order $\alpha>0$ if there exists a real constant $C$ such that
$$ d_Y(f(p),f(q)) \leq C d_X(p,q)^\alpha,\quad \text{for all}\; p,q\in X.$$
Functions which satisfy this condition are also called \emph{H{\"o}lder continuous} or \emph{$\alpha$-H{\"o}lder}. The vector space of such functions is denoted by $C^{0,\alpha}(X,Y)$ and hence $\mathrm{Lip}=C^{0,1}$.