Definition:Lipschitz Continuity

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Definition

Let $M = \struct {A, d}$ and $M' = \struct {A', d'}$ be metric spaces.

Let Let $f: A \to A'$ be a mapping.


Then $f$ is a Lipschitz continuous mapping if and only if there exists a positive real number $K \in \R_{\ge 0}$ such that:

$\forall x, y \in A: \map {d'} {\map f x, \map f y} \le K \map d {x, y}$

That is, the distance between the images of two points lies within a fixed multiple of the distance between the points.


At a Point

Let $a\in A$.


$f$ is a Lipschitz continuous at $a$ if and only if there exists a positive real number $K \in \R_{\ge 0}$ such that:

$\forall x \in A: d\,' \left({f \left({x}\right), f \left({a}\right)}\right) \le K d \left({x, a}\right)$


Lipschitz Constant

Let $f: A \to A'$ be a (Lipschitz continuous) mapping such that:

$\forall x, y \in A: d\,' \left({f \left({x}\right), f \left({y}\right)}\right) \le K d \left({x, y}\right)$

where $K \in \R_{\ge 0}$ is a positive real number.


Then $K$ is a Lipschitz constant for $f$.


Real Function

The concept can be directly applied to the real numbers considered as a metric space under the usual topology:


Let $A \subseteq \R$.

Let $f: A \to \R$ be a real function.

Let $I \subseteq A$ be a real interval on which:

$\exists K \in \R_{\ge 0}: \forall x, y \in I: \size {\map f x - \map f y} \le K \size {x - y}$


Then $f$ is Lipschitz continuous on $I$.


Also known as

A Lipschitz continuous mapping $f: A \to A'$ is also seen referred to as follows:

$f$ satisfies the Lipschitz condition on $A$
$f$ is a Lipschitz function on $A$
$f$ is Lipschitz on $A$.


Also see


Source of Name

This entry was named for Rudolf Lipschitz.


Sources