1549901369-Elements_of_Real_Analysis__Denlinger_

272 Chapter 5 • Continuous Functions

where xi, Yi E {O, 2}. Let n denote the first (smallest) natural number such
that Xn -=f. Yn. (These are the first ternary digits of x and y that are not equal.)
Then Xn = 0 and Yn = 2. Thus,

~yi/2 ~xi/2

cpc(Y) - cpc(x) = L 2i - L 2i

i=l i=l

= ~ [t Yi ; Xi + f Yi ; Xi l

i=l i=n+l

= ~ [

2

2n + f Yi; Xi]

i=n+l

1 1 00 0-2

:::: 2n + 2 I: 2i

i=n+l

1 1 2 00 1

= 2n - 2. 2n+l L 2i

i=O

1 1

= 2n - 2n+l (2)

= o.

Thus x < y ::::} cpc (y) -cpc ( x) 2:: 0 ::::} cpc ( x) ~ cpc (y). Therefore, cpc is monotone
increasing on C. •

Definition 5.5. 7 We extend cpc : C ---> [O, 1 J to a function cp : [O, 1 J ---> [O, 1 J as
follows:

For all x E [O, 1], if x E C we define cp(x) = cpc(x). If x rJ. C, then x is a
member of exactly one of the open intervals (a, b) deleted from [O, 1] to create
C ; we define cp(x) = cpc(a) = cpc(b). In summary,

{

cpc(x) if x EC; }

cp~)=.

cpc(a) if x E (a, b), where (a, b) is one of the

intervals removed from [O, 1] to create C 0

This function is called the Cantor function on [O, 1 J. It is constant on each
of the open intervals removed from [O, 1 J in creating the Cantor set. A rough
idea of its graph is sketched at the top of page 273.