Definition (Lipschitz continuous)

Let $U\subset \mathbb{R}^{n}$ be open and let $f:U\to \mathbb{R}^{n}$ be a mapping. Then f is said to be Lipschitz continuous if $\exists k>0$ s.t. $$\lVert f(x)-f(y) \rVert \le k\lVert x-y \rVert \quad\forall x,y\in U$$