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Lipschitz condition (Definition)

A mapping $ f: X \to Y$ between metric spaces is said to satisfy the Lipschitz condition, or to be Lipschitz continuous or $ L$-Lipschitz if there exists a real constant $ L$ such that

$\displaystyle d_Y(f(p),f(q)) \leq L d_X(p,q),$   for all$\displaystyle \; p,q\in X.$

The least constant $ L$ for which the previous inequality holds, is called the 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.

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
$\displaystyle d_Y(f(p),f(q)) \leq C d_X(p,q)^\alpha,$   for all$\displaystyle \; p,q\in X.$

Functions which satisfy this condition are also called Hölder continuous or $ \alpha$-Hölder. The vector space of such functions is denoted by $ C^{0,\alpha}(X,Y)$ and hence $ \mathrm{Lip}=C^{0,1}$.



"Lipschitz condition" is owned by paolini. [ full author list (3) | owner history (2) ]
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See Also: Rademacher's theorem, Newton's method, Kantorovitch's theorem

Other names:  Lipschitz, Lipschitz continuous
Also defines:  Holder, Holder continuous, Lipschitz constant

Attachments:
Lipschitz condition and differentiability (Theorem) by Mathprof
uniform continuity of Lipschitz functions (Theorem) by paolini
example of Lipschitz condition (Example) by me_and
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Cross-references: vector space, order, continuous, functions, inequality, real, metric spaces, mapping
There are 17 references to this entry.

This is version 23 of Lipschitz condition, born on 2001-11-12, modified 2005-02-28.
Object id is 765, canonical name is LipschitzCondition.
Accessed 27759 times total.

Classification:
AMS MSC: 26A16 (Real functions :: Functions of one variable :: Lipschitz classes)
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