# Definition:Lipschitz Continuity

## Contents

## 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

- 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*: $2.3$: Convex functions - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Lipschitz condition**