## The Lipschitz Condition

Definition(Lipschitz): A continuous function where is called locally Lipschitz in if for all there is a neighbourhood of and a constant such that for all we have (and in general for metrics and in metric spaces and where is a continuous map from to ).

Note that a function is locally Lipschitz in so that it obeys the Lipschitz condition on every finite interval. If we consider a function only depending on then the Definition above becomes:

A continuous function where is called  locally Lipschitz if for all there is a neighbourhood of and a constant such that for all we have It is called globally Lipschitz, if the above relation holds with the same constant  for al Example: The function is globally Lipschitz since so the condition holds with Example: The function is locally Lipschitz. Let and let be a bounded neighbourhood of  holds for all with Example: The function for where is not locally  Lipschitz. If it was, there would be a neighbourhood with and such that for all we have But then which is a contradiction if we choose small enough. Thus is not locally Lipschitz.

Lipschitz continuity is stronger than the ordinary definition of continuity, and closer to uniform continuity. It is in fact stronger than uniform continuity, since in we can take everywhere, which is fixed if is fixed. Tis implies that every Lipschitz continuous function is uniformly continous 