Definition:Lipschitz Continuity/Lipschitz Constant

Definition

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

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

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

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 Otto Sigismund Lipschitz.