Encyclopedia > Lipschitz maps

  Article Content

Lipschitz continuous

Redirected from Lipschitz maps

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. 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 : UR is a real-valued Lipschitz continuous map, then there always exist Lipschitz continuous maps MR which extend f and have the same Lipschitz constant as f.

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 zero). 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 Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Islip Terrace, New York

... Demographics As of the census of 2000, there are 5,641 people, 1,755 households, and 1,463 families residing in the town. The population density is 1,533.8/km² ...

 
 
 
This page was created in 39.3 ms