1549901369-Elements_of_Real_Analysis__Denlinger_

164 Chapter 3 • Topology of the Real Number System

12. Let f : V(f) --+ JR be a function with domain V(f), and let A ~ V(f).

3.4

We say that f is bounded on A if :JB > 0 3 Vx EA, lf(x)I ::; B. We

say that f is locally bounded at a point x if 3 8 > 0 3 f is bounded on

Na(x).

(a) Prove that if f is locally bounded at every point of a compact set A,

then f is bounded on A.

(b) Find a function f that is locally bounded at every point of (0, 1) but

not bounded on (0, 1).

• The Cantor Set

The Cantor set is a most remarkable set of real numbers. It is a subset of [O, l]
obtained by removing, in successive steps, a sequence of subsets of [O, l]. Ill
some sense, it is easier to visualize the complement of the Cantor set than it is
to visualize the set itself; the Cantor set is what is "left over" after the removal
process.

Definition 3.4.1 The Cantor Set:

Let Co = [O, l] and C 1 = the set remaining after removing ( ~, ~), the "open
middle third" of Co. Thus,

0 1 2 1

3 3

Figure 3.15

Similarly, let C2 = the set remaining after removing the "open middle
thirds" ( i , ~) and ( ~, ~) , each of length i, from the intervals comprising C 1.

[

0 1 2

9 9

I

3

Figure 3.16

2

3

]

7 8

9 9