“real: chapter_05” — 2011/5/22 — 22:55 — page 29 — #29
Limits and Continuity 5-29
uniformly continuous. Consider any function f : Z → R, where Z is
the set of all integers.
In conclusion, we note that there are unbounded subsets of
R on
which every function is uniformly continuous. The following example
illustrates this.
Example 5.6.7 Let S =
Z, the set of all integers. Every function
f :
Z → R is not only continuous but is uniformly continuous by
taking δ<1 for every >0.
Theorem 5.6.8 Let f : S ⊂
R → R be uniformly continuous. If x
0
is
a limit point of S then f has a limit at x
0
.
Proof Let x
0
be a limit point of S and let {x
n
} be any sequence in
S \{x
o
} converging to x
0
. In view of Theorem 5.2.11, it is sufficient
to prove that the sequence {f (x
n
)} is Cauchy ...