Lipschitz continuity

In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M. In this case, K is called the Lipschitz constant of the map. The name comes from the German mathematician Rudolf Lipschitz.

Every Lipschitz continuous map is uniformly continuous and hence continuous.

Lipschitz continuous maps with Lipschitz constant K=1 are called short maps and with K < 1 are called contraction mappings when M=N also; the latter are the subject of the Banach fixed point theorem.

Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.

If U is a subset of the metric space M and f : U → R is a real-valued Lipschitz continuous map, then there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem).

A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure zero). If K is the Lipschitz constant of f, then |f'(x)| ≤ K whenever the derivative exists. Conversely, if f : I → R is a differentiable map with bounded derivative, |f'(x)| ≤ L for all x in I, then f is Lipschitz continuous with Lipschitz constant K ≤ L, a consequence of the mean value theorem.