290 Chapter 5 • Continuous Functions- 7 *sets of Points of Discontinuity (Project)
This section is icing on the cake. It answers several intriguing
questions but is not needed in any of the later sections of the
book, except in certain advanced exercises. It is cast in the
form of a project.In this section, we shed some light on the question of what kind of set can pos-
sibly be the set of points where a given function f : 7J(f) --+ JR is discontinuous.
(We call such a set the "set of discontinuities" of f.) Exploring this question
will lead to some interesting, even surprising, results. First, a routine result.
Theorem 5.7.1 Given any finite set A = {a 1 ,a2, ···, an} of real numbers,
there is a bounded function f : JR --+ JR having the set A as its set of disconti-
nuities.
Proof. Exercise. [Hint: Think of a characteristic function.] •As we have seen , the set of discontinuities of a function can be infinite.
For example, the Dirichlet function (5.1.11) is discontinuous everywhere, and
Thomae's function (5.1.12) is discontinuous on the rational numbers. The latt er
example leads us to ask whether the set of rational numbers is special in this
regard, or whether any countable set can be the set of discontinuities of a
function. Here is the answer to the second part of that question.
Theorem 5. 7.2 Given any countable set A of real numbers, there is a bounded
function f : JR --+ JR having A as its set of discontinuities.
Proof. Exercise. Here is a recommended approach. Let A be a countable
set. The finite case is covered in Theorem 5.7.l. Thus, assume
A= {a1, a2, · · · , an,···} is infinite. Define f: JR--+ JR by f(x) =
{.!. n if x = an. E A }. ,,.., .LO s h ow t h at f 1s. d. IScontmuous. on A , s h ow t h at Ac. is
0 otherwise
dense in JR and follow the argument given in the proof of part (b) of Theorem
5.1.12. To show that f is continuous on Ac, let x 0 E Ac. Then f(x 0 ) = 0. Let
c > 0. Then 3 n E N 3 .!. < €. Then there is a neighborhood N of x 0 such that
ai tJ. N, a2 tJ. N, · · · , an ntf. N. Then x EN=> 0 :S: f(x) < ~ < €. •
While the above result is remarkable, here is another, even more amazing
result.
Theorem 5.7.3 Given any countable set A of real numbers, there is a bounded
monotone increasing function f : JR --+ JR having A as its set of discontinuities.