Lipschitz continuous

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 contant K < 1 are called contractions; they 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.

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.