Definition:Lipschitz Continuity/Point

Definition

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

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

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:

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

Sources

• 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control: $2.3$: Convex functions