Hölder continuity - Imported Knowlpedia Content

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

d_Y\bigl(f(x),f(y)\bigr)\le C\,\bigl(d_X(x,y)\bigr)^{\alpha} \quad\text{for all } x,y\in X.

Remarks

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 $\alpha$ .