Locally Lipschitz function

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.