Lipschitz Continuity

Definition

A function $f:X\to Y$ is $k$-Lipschitz continuous, when $\exists k\ge 0\in \mathbb{R}$ such that $|f(x_1)-f(x_2)|\le k|x_1-x_2|$ where $x_1,x_2\in X$

A strong form of uniform continuity for functions.

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 $0<0\le 1$