Lipschitz continuity

A function $f$ is Lipschitz continuous when there is a constant $L\ge 0$ such thatfor all points in its domain.

A one-sided Lipschitz condition controls the growth of distances between solutions of an ordinary differential equation. Differentiating the squared difference and applying the Gronwall inequality gives $\Vert y(t)-z(t)\Vert \le e^{\nu t}\Vert y(0)-z(0)\Vert$. The case $\nu \le 0$ is a dissipative vector field. Ordinary Lipschitz continuity implies such a bound, but the converse fails: $f(y)=-y^3$ has one-sided constant zero on the real line without being globally Lipschitz continuous.

A Lipschitz domain is an open connected subset of Euclidean space whose boundary is locally the graph of a Lipschitz continuous function after rotating and translating coordinates. Locally the domain lies on one side of that graph. This permits corners but excludes boundary cusps that cannot be represented with a finite Lipschitz constant. Bounded Lipschitz domains are a standard setting for Sobolev spaces and the Poincaré inequality for total variation.

If $f:A\to \mathbb{R}$ is $L$-Lipschitz on a subset of a metric space, thenextends $f$ to the whole space and remains $L$-Lipschitz.

The least $L\ge 0$ for which $d(f(x),f(y))\le Ld(x,y)$ for every $x,y$ is the Lipschitz constant of $f$. Any admissible $L$ is a Lipschitz bound.

A function is globally Lipschitz when one Lipschitz bound holds for every pair of points in its domain. Unlike local Lipschitz continuity, this condition gives a uniform growth control over the whole domain.

A Lipschitz bound has the form $|f(x)-f(y)|\le L|x-y|$. It controls the change of a function by the change of its argument.

A function is locally Lipschitz when every point has a neighborhood on which the function satisfies a Lipschitz bound. Continuously differentiable functions are locally Lipschitz.