Hölder continuity

A Hölder continuous map between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is a map $f\colon X\to Y$ such that there exist constants $C\ge 0$ and $\alpha\in(0,1]$ with

When $\alpha=1$, Hölder continuity is exactly Lipschitz continuity . For any $\alpha\in(0,1]$, Hölder continuity implies uniform continuity .

Examples:

• The function $f(x)=\sqrt{x}$ on $[0,1]$ is Hölder continuous with exponent $\alpha=\tfrac12$.
• For $\alpha\in(0,1]$, the function $f(x)=|x|^{\alpha}$ on $\mathbb{R}$ is Hölder continuous with exponent $\alpha$.