Compactness is intimately connected with the properties of continuity, uniform continuity, closedness and boundedness. One such is illustrated in the following theorem.
Theorem
Let
be continuous with
compact (closed and bounded. Then
is uniformly continuous.
Proof: Choose
Sinceis continuous on
is continuous at
for each
Thus for each
there is
such that if
and
then
Consider the family of intervals
This is an open cover ofandis compact so there is a finite subcover of
so there are
such that
Let
and suppose
and
then there is
such that
Now
making
Hence
andis uniformly continuous.
This theorem has a converse. Supposeis uniformly continuous thenhas a limit at each accumulation point of
Let
be the set of accumulation points ofand let define
Define
by
forand
for 
The function
is continuous.