Definition(Lipschitz) A continuous function
where
is called locally Lipschitz in
if for all
there is a neighbourhood
ofand a constant
such that for all
we have
(and in general
for metrics
and
in metric spaces
and
where 
is a continuous map fromto).
Note that a
functionis 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 allthere is a neighbourhoodofand a constantsuch 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 letbe a bounded neighbourhood of
holds for allwith
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 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 in
we can take
everywhere, which is fixed if
is fixed. This implies that every Lipschitz continuous function is uniformly continuous