= Locally Lipschitz function
= Locally Lipschitz
{synonym}
A <function> between <metric spaces> is locally Lipschitz if every point has a <neighbourhood> on which one finite <Lipschitz bound> works for every pair. The neighbourhood and bound may depend on the point. A <continuously differentiable> real function on an open <interval> has this property: continuity of its <derivative> bounds that derivative on a sufficiently small closed interval, and the <mean value theorem> bounds every secant slope there.
The <function> $x^2$ is locally Lipschitz at every real point but has no single bound on the <real line>. The <square root> is locally Lipschitz on $(0,\infty)$, and has the bound $1/(2\sqrt a)$ on $[a,b]$ for $0<a<b$. Adding zero to its domain changes the conclusion at that point: the ratios in the <Lipschitz continuity> example become unbounded, so it is not locally Lipschitz at zero. This distinction is what permits the delayed, nonunique ODE solutions illustrated under <Picard-Lindelöf theorem>.
Back to article page
