Lipschitz continuity

(Redirected from Lipschitz continuous)

In mathematics, a function

f : MN

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. The smallest such K is called the Lipschitz constant of the map. The name is for the German mathematician Rudolf Lipschitz.

Every Lipschitz continuous map is uniformly continuous, and hence a fortiori 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 : UR is a Lipschitz continuous map, there always exist Lipschitz continuous maps MR which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem).

A Lipschitz continuous map f : IR, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0). If K is the Lipschitz constant of f, then |f'(x)| ≤ K whenever the derivative exists. Conversely, if f : IR is a differentiable map with bounded derivative, |f'(x)| ≤ L for all x in I, then f is Lipschitz continuous with Lipschitz constant KL, a consequence of the mean value theorem.

All Banach spaces have the notion of Lipschitz continuity.

Hölder continuity

If a map fM → N satisfies the Lipschitz-like condition

d(f(x), f(y)) ≤ Kd(xy)α

for some α > 0 (the order) and all x, y, it is said to be Hölder-continuous or α-Hölder.

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