Lipschitz condition

A mapping f:XY between metric spaces is said to satisfy the Lipschitz conditionMathworldPlanetmath, or to be Lipschitz continuous or L-Lipschitz if there exists a real constant L such that

dY(f(p),f(q))LdX(p,q),for allp,qX.

The least constant L for which the previous inequalityMathworldPlanetmath holds, is called the Lipschitz constant of f. The space of Lipschitz continuous functions is often denoted by Lip(X,Y).

Clearly, every Lipschitz continuous function is continuousMathworldPlanetmath.


More generally, one says that a mapping satisfies a Lipschitz condition of order α>0 if there exists a real constant C such that

dY(f(p),f(q))CdX(p,q)α,for allp,qX.

Functions which satisfy this condition are also called Hölder continuous or α-Hölder. The vector space of such functions is denoted by C0,α(X,Y) and hence Lip=C0,1.

Title Lipschitz condition
Canonical name LipschitzCondition
Date of creation 2013-03-22 11:57:48
Last modified on 2013-03-22 11:57:48
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 27
Author paolini (1187)
Entry type Definition
Classification msc 26A16
Synonym Lipschitz
Synonym Lipschitz continuous
Related topic RademachersTheorem
Related topic NewtonsMethod
Related topic KantorovitchsTheorem
Defines Holder
Defines Holder continuous
Defines Lipschitz constant