Lipschitz continuity
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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.
Hölder continuity
If a map f: M → N satisfies the Lipschitz-like condition
- d(f(x), f(y)) ≤ Kd(x, y)α
for some α > 0 (the order) and all x, y, it is said to be Hölder-continuous or α-Hölder.
See also
es:Lipschitz continua fr:Application lipschitzienne pl:Warunek Lipschitza