6.7 Uniform Continuity
There is a theorem about the integral of a continuous function which requires the notion of
uniform continuity. This is discussed in this section. Consider the function f
This is a continuous function because, by Theorem 6.0.6
, it is continuous at every
However, for a given ε >
needed in the ε,δ
definition of continuity
becomes very small as x
gets close to 0. The notion of uniform continuity involves being able
to choose a single δ
which works on the whole domain of f.
Here is the definition.
Definition 6.7.1 Let f be a function. Then f is uniformly continuous if for
every ε > 0, there exists a δ depending only on ε such that if
< δ then
It is an amazing fact that under certain conditions continuity implies uniform continuity.
Theorem 6.7.2 Let f : K → F be continuous where K is a sequentially
compact set in F. Then f is uniformly continuous on K.
Proof: If this is not true, there exists ε > 0 such that for every δ > 0 there exists a pair of
points, xδ and yδ such that even though
succession of values for δ
equal to 1,
and letting the exceptional pair of points for
be denoted by xn
Now since K is sequentially compact, there exists a subsequence,
xnk → z ∈ K.
Now nk ≥ k
Consequently, ynk → z also. ( xnk is like a person walking toward a certain point and ynk is
like a dog on a leash which is constantly getting shorter. Obviously ynk must also move
toward the point also. You should give a precise proof of what is needed here.) By continuity
of f and Theorem 6.1.2,
an obvious contradiction. Therefore, the theorem must be true. ■
The following corollary follows from this theorem and Theorem 4.7.2.
Corollary 6.7.3 Suppose K is a closed interval,
, a set of the form
Then f is uniformly continuous.