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

$$d_Y(f(p),f(q))\le L{d}_X(p,q),\text{for all}p,q\in X.$$

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

$$d_Y(f(p),f(q))\le C{d}_X{(p,q)}^{\alpha },\text{for all}p,q\in X.$$

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