Definition:Hölder Continuous

Definition

Let $\left({M_1, d_1}\right)$ and $\left({M_2, d_2}\right)$ be metric spaces.

Let $\alpha \in \R_{\ge 0}$ be a positive real number.

A mapping $f: M_1 \to M_2$ is said to be $\alpha$-Hölder continuous if and only if:

$\exists L \in \R_{\ge 0}: \forall x, y \in M_1: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) \le L \left({d_1 \left({x, y}\right)}\right)^\alpha$

Further, $f$ is said to be Hölder continuous if and only if it is $\alpha$-Hölder continuous for some $\alpha \in \R_{\ge 0}$.

Source of Name

This entry was named for Otto Ludwig Hölder.