The Real And Complex Number Systems

(b) Note that since fS R, andfS is connected, we know that fS is an interval I.
Given a S, then fa I. We discuss 2 cases as follows. (1) fa is an interior point of I.
(2) fa is the endpoint of I.
For case (1), it is similar to (a). We omit the proof.
For case (2), it is similar to (a). We omit the proof.
So, we have proved that f is continuous on S.
(c) Given a S, then fa fS. SincefS is closed, we consider two cases. (1) fa
is an isolated point and (2) fa is an accumulation point.
For case (1), claim that a is an isolated point. Suppose NOT, there is a sequence
x n S with x n a. Consider x n n1
x : x n a x : x n a, and thus we
may assume that x : x n a : a n is a infinite subset of x n
n1
.Sincef is
monotonic, we have lim n fx n fa . SincefS is closed, we have fa fS.
Therefore, there exists b fS such that fa fb fa.
If fb fa, then b a since f is strictly increasing. But is contradicts to that fa is
isolated. On the other hand, if fb fa, then b a since f is strictly increasing. In
addition, fa n fa fb implies that a n b. But is contradicts to that a n a.
Hence, we have proved that a is an isolated point. So, f is sutomatically continuous at
a.
For case (2), suppose that fa is an accumulation point. Then Bfa; fS and
Bfa; has infinite many numbers of points in fS. Choose y 1 , y 2 Bfa; fS
with y 1 y 2 , then fx 1 y 1 ,andfx 2 y 2 . And thus it is similar to (a), we omit the
proof.
So, we have proved that f is continuous on S by (1) and (2).
Remark: In (b), when we say f is monotonic on a subset of R, its image is also in R.
Supplement.
It should be noted that the discontinuities of a monotonic function need not be isolated.
In fact, given any countable subset E of a, b, which may even be dense, we can
construct a function f, monotonic on a, b, discontinuous at every point of E, and at
no other point of a, b. To show this, let the points of E be arranged in a sequence x n ,
n 1, 2, . . . Let c n be a sequence of positive numbers such that c n converges. Define
fx c n a x b
x nx
Note: The summation is to be understood as follows: Sum over those indices n for whcih
x n x. If there are no points x n to the left of x, the sum is empty; following the usual
convention, we define it to be zero. Since absolute convergence, the order in which the
terms are arranged is immaterial.
Then fx is desired.
The proof that we omit; the reader should see the book, Principles of Mathematical
Analysis written by Walter Rudin, pp97.
Metric space and fixed points
4.66 Let BS denote the set of all real-valued functions which are defined and
bounded on a nonempty set S. Iff BS, let
f sup
xS
The number f is called the " sup norm " of f.
|fx|.