Lipschitz condition

Definition

Let $S$ be a subset of $\mathbb{R}^n$, let $f : S \to \mathbb{R}^m$ be a function, and let $K \ge 0$ be a real number. The function $f$ satisfies a Lipschitz condition on $S$ with constant $K$, or is $K$-Lipschitz on $S$, if $$\| f(x) - f(y) \| \le K\, \| x - y \| \quad \text{for all } x, y \in S.$$ Let $D \subseteq \mathbb{R} \times \mathbb{R}^n$ and $f : D \to \mathbb{R}^m$. The function $f$ is Lipschitz in $x$ on $D$ with constant $K$ if $\| f(t, x) - f(t, y) \| \le K \| x - y \|$ whenever $(t, x), (t, y) \in D$.

Remarks

A $K$-Lipschitz function is uniformly continuous. A differentiable function on a convex set with $\| f'(x) \| \le K$ at each point is $K$-Lipschitz, by the mean value inequality. For example, $\sin$ is $1$-Lipschitz on $\mathbb{R}$. The function $x \mapsto \sqrt{|x|}$ is continuous but not Lipschitz on any interval that contains $0$. The non-uniqueness example uses a function of this kind. The Picard–Lindelöf theorem needs a Lipschitz condition in $x$.

Explanations

L1 human

A Lipschitz function cannot change faster than a fixed rate. The slope of each chord of its graph is at most $K$ in absolute value.