1549901369-Elements_of_Real_Analysis__Denlinger_

5.5 *Monotonicity, Continuity, and Inverses 271

REMARKS:

(1) 'Pc(x) is the binary expansion of a unique real number in [O, l].

(2) 'Pc is onto [O, l]. This is because C consists of all possible ternary

expansions of real numbers in [O, 1] that consist of only O's and 2's, implying

that the range of 'Pc consists of all possible binary expansions of real numbers

in [O, l].

(3) 'Pc ( ~) = 'Pc ( ~) = ~. This is because, in base-3,

~ = 0.02222222 · · · , and

~ = 0.20000000 ... ,

while in base-2, 0.011111111 · · · = 0.100000000 = ~-

( 4) 'Pc ( ~) = 'Pc ( ~) = i, because, in base-3,

~ = 0.002222222 · · · , and

~ = 0.020000000 ... ,

while in base-2, 0.0011111111 · · · = 0.0100000000 = i· Also,

'Pc(~) ='Pc(~) =~,because, in base-3,

l 9 = ±. 3 + l 3 = O. 202222222 · · · , and

~ = ~ + ~ = 0.220000000 ... ,

while in base-2, 0.1011111111 · · · = 0.1100000000 = ~-

(5) By remark (3), 'Pc takes on the same value at both end points of the

interval "removed" to create C 1 (see Definition 3.4.1). By remark (4), 'Pc takes

on the same value at both endpoints of each interval "removed" to create C 2.

(6) Continuing in this way, we can show that 'Pc takes on the same value
at both endpoints of every interval "removed" to create the Cantor set C. (We
omit the details.)

Theorem 5.5.6 'Pc : C--> [O, 1] is monotone increasing on C.

Proof. Let x <yin C. Expressing x and yin ternary decimal form,

00 00

x = '""X~ 3i i and y = '""Yi ~ 3i ,

i=l i=l