1549901369-Elements_of_Real_Analysis__Denlinger_

270 Chapter 5 • Continuous Functions

while

and

c < x < d ::::} f(x) lies between f(c) and f(d)

* f(d) < f(x) < f(c)

x>d* x>d>c

::::} f(d) lies between f(x) and f(c)

* f(x) < f(d) < f(c).

From these inequalities we see that 't/x E I ,

{

x < c::::} f(x) > f(c) }·

x > c::::} f(x) < f(c)

(13)

Now, consider any x 1 < x 2 in I. We have the following cases to consider:
Case 1 (x 1 :::; c:::; x 2 ): Then by (13) and the 1-1 property off, we have
f(x1) > f(x2).

Case 2 (x 1 < x2 < c): Then by (13) we have f(x1) > f(c) and by Claim

#1, f(x2) must lie between f(x1) and J(c), so f(x2) < f(x1).

Case 3 (c < x 1 < x2): Then by (13) we have f(x2) < f(c) and by Claim

#1, f(x1) must lie between f(x2) and f(c), so f(x1) > f(x2).

In all cases x 1 < x2 =? f(x 1 ) > f(x2). That is, f is strictly decreasing on I.
Therefore, f is either strictly increasing or strictly decreasing on I. •

THE CANTOR FUNCTION

The Cantor set described in Section 3.4 allows us to define a curious func-
tion, called the "Cantor function,'' cp : [O, 1] --r [O, 1], which is continuous and
monotone increasing; f (0) = 0, f (1) = 1, and yet f is not strictly increasing
on any nonempty open interval. This function will be defined in stages, first on
the Cantor set C , and then extended to [O, 1].

Definition 5.5.5 Defining the function cp, first on the Cantor set.

Recall that the Cantor set C consists of all those real numbers x in [O, 1]
with a ternary (base-3) decimal expansion consisting of only O's and 2's:

00

X= L;:.

i=l

Define 'Pc : C --t [O, 1] by

~xi/2

'Pc(x) = ~ Zi'

i=l

(where Xi = 0 or 2)

regarded as a binary (base-two) decimal.