1549901369-Elements_of_Real_Analysis__Denlinger_

5.7 *sets of Points of Discontinuity (Project) 293

Definition 5. 7.12 A set is called an F,, set if it is the union of countably many
closed sets.^20

Comments: An F,, set is not necessarily closed, as shown by Example
3.2.5. Theorem 5.7.11 shows that the set of discontinuities of a function f :
D(f) -. JR must be an F,, set. We now see that our detour brought us some
new insight into sets of discontinuity. To show that not every set of real numbers
can be the set of discontinuities of a function we need to show that not every
set of real numbers is an F,, set. Before doing this we introduce a few technical
results.

FIRST AND SECOND CATEGORY SETS

Definition 5. 7.13 Let us call an interval a proper interval if it is of the form
(a,b), (a,b], [a,b), or [a,b], where a< b.

Recall that a set A is nowhere dense if its closure A contains no proper
intervals (see Definition 3.4.16). Thus, a closed set either contains a proper
interval or is nowhere dense.

Lemma 5. 7.14 Suppose A is a nowhere dense set. Then, for every proper
interval I, there is a proper closed interval J 5;:;; I such that Jn A= 0.

Proof. Suppose A is a nowhere dense set. Let I be a proper interval, say
(a, b) 5;:;; I 5;:;; [a , b], where a < b. Since A is nowhere dense, (a, b) </:. A, so
3 x E (a, b) -A. Since x E Ac, which is open, 3 8 > 0 such that (x - 8, x + 8) 5;:;;
Ac. Choosing 8' sufficiently small, we have [x - 81 ,x + 8'] 5;:;; Ac n J. Take
J = [x - 81 , x + 8']. •

I

N 0 (x)

[ ( I ) ]

a x b

CAc

(open)

Figure 5.13

Definition 5. 7.15 A set of real numbers is said to be of the first category
if it is the union of a countable collection of nowhere dense sets; otherwise, it
is said to be of the second category.

20. F is the first letter of the French word for "closed," and a is the Greek equivalent of the
    first letter of "sum," for union.