5.5 *Monotonicity, Continuity, and Inverses 2 7 3y 1 7 8 3 4 5 8 I 2 3 8 I 4 l 8I 2 I 2 7 8 x(^9 9 3 3 9 9)
Fig ure 5.1 2
Theorem 5.5.8 The Cantor function cp : [O, 1]
0
~
0
[O, 1] is continuous and
monotone increasing, yet is 'not strictly increasing on any nonempty open in-
terval.
Proof. We have seen by construction that cp is monotone increasing, and
that cp[O, 1] = [O, 1]. Thus, Theorem 5.5.2 tells us that cp is continuous. Also,
by construction cp is constant on each of the disjoint open intervals comprising
the complement of the Cantor set.
Let I denote any nonempty open subinterval of [O, 1]. Since C contains no
nonempty open intervals (Theorem 3.4.3) I must contain a point xi C. Then,
using de Morgan's law,
00 00
x E [O, 1] - n C n = LJ ([O, 1] - C n)·
n=l n=l
T hus, x belongs to one of the open intervals J that was removed to create
C. Since I and J are open, and x EI n J, =i c: > 0 3
Then cp is constant on Ne(x) since it is constant on J. Thus, cp is not strictly
increasing on Ne(x). Therefore, cp is not strictly increasing on I. •