Lipschitz condition

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

 $d_{Y}(f(p),f(q))\leq Ld_{X}(p,q),\quad\text{for all}\;p,q\in X.$

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

 $d_{Y}(f(p),f(q))\leq Cd_{X}(p,q)^{\alpha},\quad\text{for all}\;p,q\in X.$

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 $\mathrm{Lip}=C^{0,1}$.

 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