Definition:Lipschitz Continuity/Point

Definition

Let $M = \left({A, d}\right)$ and $M' = \left({A', d\,'}\right)$ 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:

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

Sources

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