Definition:Hölder Continuous
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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 $\alpha \in \R_{\ge 0}$.
Also see
- Lipschitz Continuity: the case where $\alpha = 1$
Source of Name
This entry was named for Otto Ludwig Hölder.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $15.1$