Holder Continuous

Definition

A function $f:X\to Y$ is $\alpha$-Holder continuous $f\in C^{\alpha }$, when $\exists C\ge 0\in \mathbb{R}$ such that $|f(x)-f(y)|\le C||x-y|{|}^{\alpha }$ where $x,y\in X$

Facts

Continuously differentiable $\subset$ Lipschitz continuous $\subset$ $\alpha -$Holder Continuous $\subset$ Uniformly Continuous $\subset$ Continuous Lipschitz continuous Absolute continuous $\subset$ Uniformly Continuous $\subset$ Continuous where $0<0\le 1$

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