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. 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 : U → R is a Lipschitz continuous map, 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 0). 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. All Banach spaces have the notion of Lipschitz continuity.

Hlder continuity

If a map

f:M \to N

satisfies the Lipschitz-like condition

d(f(x),f(y)) \le K d(x,y)^\alpha

for some

\alpha > 0

(the order) and all x, y, it is said to be Hlder-continuous or $\alpha$ -Hlder.

Lipschitz Continuity

Hlder continuity

See also