Unformatted text preview: e that the limit of x n belongs to S.
In view of Part a, the present result follows at once from an earlier theorem on limits of
d. Suppose that f is uniformly continuous on a set S, that x is a real number and that x n and t n are
sequences in S that converge to the number x. Prove that
lim f x n n Ý f t n .
The existence of these limits was guaranteed in Part c. Now since t n x n 0 as n Ý we
deduce from the relationship between uniform continuity and limits of sequences that
0 as n Ý.
e. Suppose that f is uniformly continuous on a set S and that x S S. Explain how we can use Part d to
extend the definition of the function f to the number x in such a way that f is continuous on the set
SÞ x .
We know that there exists a sequence x n in S that converges to x and we know that there is a
single limit for all of the sequences f x n that can be made in this way. We define f x to be
this common limit value. This extension of the function f to the set S Þ x is uniformly
continuous. The proof will be given in the more extended case that we consider below in Part f.
f. Prove that if f is uniformly continuous on a set S then it is possible to extend f to the closure S of S in
such a way that f is uniformly continuous on S.
For every number x S S we define f x by the method described in Part e. To show that the
extension of f to S is uniformly continuous, suppose that 0. Using the uniform continuity of f
on S we choose 0 such that the inequality
|f t f x |
holds whenever t and x belong to S and |t x | . We shall now observe that whenever t and x
belong to S and |t x | we must have |f t f x | . To make this observation, suppose that
t and x belong to S and that |t x | .
choose a sequence t n in S that converges to t and a sequence x n in S that converges to x.
|t x | n Ý|t n x n |
we know that the inequality |t n x n | must hold for all n sufficiently large and therefore, since 204 |f t f x | n Ý|f t n
lim f xn | and since
|f t n
for all n sufficiently large we must have
|f t f xn |
fx | 2 2
. 10. Suppose that f is a continuous function on a bounded set S. rove that the following two conditions are
a. The function f is uniformly continuous on S.
b. It is possible to extend f to a continuous function on the set S.
The fact that condition a implies condition b follows from Exercise 9. On the other hand, if f has a
continuous extension to the set S then, this extension, being a continuous function on a closed
bounded set, must be uniformly continuous; and so f must be uniformly continuous on S.
11. Given that f is a function defined on a set S of real numbers, prove that the following conditions are
a. The function f fails to be uniformly continuous on the set S.
b. There exists a number
n Ý and 0 and there exist two sequences t n and x n in S such that t n
|f x n xn 0 as f tn | for every n.
At the suggestion of my good friend Sean Ellermeyer this exerci...
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