A map 
from a metric space 
to a metric space 
is said to be uniformly continuous if for every 
, there exists a 
such that 
whenever 
satisfy 
.
Note that the 
here depends on 
and on 
but that it is entirely independent of the points 
and 
. In this way, uniform continuity is stronger than continuity
and so it follows immediately that every uniformly continuous function is continuous.
Examples of uniformly continuous functions include Lipschitz functions and those satisfying the Hölder
condition. Note however that not all continuous functions are uniformly continuous
with two very basic counterexamples being 
(for 
) and 
(for 
. On the other hand, every function which is continuous
on a compact domain is necessarily uniformly continuous.
