# Lipschitz continuous

The function $LaTeX: f: X \rightarrow \mathbb{R}$ is Lipschitz continuous if there exists a value $LaTeX: K$, called the Lipschitz constant, such that $LaTeX: |f(x) - f(y)| \le K \|x-y\|$ for all $LaTeX: x$ and $LaTeX: y$ in $LaTeX: X$. This relation is called the Lipschitz condition. It is stronger than continuity because it limits the slope to be within $LaTeX: [-K, K]$. The generalized Lipschitz condition is that there exists a monotonically increasing function, $LaTeX: M : \mathbb{R} \rightarrow \mathbb{R}$ with the property that $LaTeX: m(z) \rightarrow 0$ as $LaTeX: z \rightarrow 0$ such that there exists $LaTeX: K$ for which $LaTeX: |f(x)-f(y)| \le K m(\|x-y\|)$ for all $LaTeX: x$ and $LaTeX: y$ in $LaTeX: X.$