The Real And Complex Number Systems

clA E E
clD C 2 A B E
clD C 2 B E
clD C 2 E since B D
clC 2 E since clD E
clC 2 C 1 since E C 1
.
Exercise Prove that every connected metric space with at least two points is uncountable.
Proof: LetX be a connected metric space with two points a and b, where
a b. Define a set A r x : dx, a r and B r x : dx, a r. It is clear
that both of sets are open and disjoint. Assume X is countable. Let
da,b
r , da,b , it guarantee that both of sets are non-empty. Since
4 2
da,b
, da,b is uncountable, we know that there is a 0 such that
4 2
A B X. It implies that X is disconnected. So, we know that such X is
countable.
Uniform continuity
4.50 Prove that a function which is uniformly continuous on S is also continuous on S.
Proof: Let f be uniformly continuous on S, thengiven 0, there exists a 0 such
that as dx, y , x and y in S, then we have
dfx, fy .
Fix y, called a. Thengiven 0, there exists a 0 such that as dx, a , x in S,
then we have
dfx, fa .
That is, f is continuous at a. Sincea is arbitrary, we know that f is continuous on S.
4.51 If fx x 2 for x in R, prove that f is not uniformly continuous on R.
Proof: Assume that f is uniformly continuous on R, thengiven 1, there exists a
0 such that as |x y| , wehave
|fx fy| 1.
Choose x y ,(
2
|x y| , then we have
2
|fx fy| y 1. 2
When we choose y 1 , then 1
2
2
1
2
1
which is absurb. Hence, we know that f is not uniformly continuous on R.
Remark: There are some similar questions written below.
1. Here is a useful lemma to make sure that a function is uniformly continuous on
a, b, but we need its differentiability.
(Lemma) Letf : a, b R R be differentiable and |f x| M for all x a, b.
Then f is uniformly continuous on a, b, wherea, b may be .