# Definition:Lipschitz Continuity/Lipschitz Constant

## Definition

Let $M = \left({A, d}\right)$ and $M' = \left({A', d\,'}\right)$ be metric spaces.

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 $f$.

## Also defined as

Some sources define the Lipschitz constant for $f$ as being the smallest $K \in \R_{>0}$ for which $f$ is Lipschitz continuous.

## Source of Name

This entry was named for Rudolf Lipschitz.