Lipschitz continuity

• Definition
  • Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, where $d_X$ denotes the metric on the set $X$ and $d_Y$ is the metric on set $Y$, a function $f : X → Y$ is called Lipschitz continuous if there exists a real constant $K ≥ 0$ such that, for all $x_1$ and $x_2$ in $X$, $$ d_Y(f(x_1), f(x_2)) \leq K \cdot d_X(x_1, x_2).$$
  • In particular, a real-valued function $f : \mathbb{R} → \mathbb{R}$ is called Lipschitz continuous if there exists a positive real constant $K$ such that, for all real $x_1$ and $x_2$, $$ |f(x_1) - f(x_2)| \leq K \cdot |x_1 - x_2|. $$