The Lipshitz Condition
Definition(Lipschitz) A continuous functionwhere is called locally Lipschitz inif for allthere is a neighbourhoodofand a constantsuch that for allwe have(and in generalfor metricsandin metric spacesandwhere is a continuous map fromto).
Note that afunctionis locally Lipschitz inso that it obeys the Lipschitz condition on every finite interval. If we consider a function only depending onthen the Definition above becomes:
A continuous functionwhereis called locally Lipschitz if for allthere is a neighbourhoodofand a constantsuch that for allwe have
It is called globally Lipschitz, if the above relation holds with the same constant for al
Example: The functionis globally Lipschitz sinceso the condition holds with
Example: The functionis locally Lipschitz. Letand letbe a bounded neighbourhood of
holds for allwith
Example: The functionforwhereis not locally Lipschitz. If it was, there would be a neighbourhoodwithandsuch that for all we have
But thenwhich is a contradiction if we choosesmall enough. Thusis 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 inwe can takeeverywhere, which is fixed ifis fixed. This implies that every Lipschitz continuous function is uniformly continuous