Hölder continuity

Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, let $f \colon X \to Y$ be a function, and let $C \ge 0$ and $\alpha > 0$ be real numbers. The map $f$ is Hölder continuous with constant $C$ and exponent $\alpha$ when $$d_Y\bigl(f(x), f(y)\bigr) \le C \, d_X(x, y)^{\alpha} \quad \text{for all } x, y \in X.$$

Remarks

For $\alpha = 1$, this is the condition for a Lipschitz map with constant $C$. The map $x \mapsto \sqrt{x}$ on $[0, \infty)$ is Hölder continuous with $C = 1$ and $\alpha = 1/2$. A Hölder continuous map is uniformly continuous: for $\varepsilon > 0$ and $C > 0$, take $\delta = (\varepsilon / C)^{1/\alpha}$. For a real function on an interval, Hölder continuity with an exponent $\alpha > 1$ forces the function to be constant.

Explanations

L1 human

The Hölder condition is a weaker form of the Lipschitz condition. It allows a map to be steep, as $\sqrt{x}$ is near $0$, but it still controls how fast the values can change.