5.7 *sets of Points of Discontinuity (Project) 295Lemma 5.7.18(a) A closed set either contains a proper interval or is nowhere dense.(b) An Fa set either contains a proper interval or is of the first category.Proof. Exercise. •Corollary 5.7.19 The set of irrational numbers is not an Fa set.At last, we can prove our desired result.Corollary 5. 7.20 There does not exist a function f : V(f) ---+ JR having the
set of irrational numbers as its set of discontinuities.Proof. Explain this. •Corollary 5.7.20 is truly remarkable! We can only marvel at the fact that
there are functions continuous on the irrationals and discontinuous on the ratio-
nals, but there are NO functions continuous on the rationals and discontinuous
on the irrationals!
Finally, the situation described in Theorem 5.7.11 is really an if-and-only-if
condition, as expressed in the following theorem.
Theorem 5. 7.21 Let A ~ R Then there exists a function f : JR ---+ JR having
A as its set of discontinuities if and only if A is an Fa set.
Proof. The "=?" direction was proved in Theorem 5.7.11. For a proof of
the "~" direction, see Gelbaum and Olmsted [49], Section 2, # 23. •
Example 5. 7.22 Given any closed set A , there is a function f : JR---+ JR having
A as its set of discontinuities.
Proof. Consider the following function:^21f ( x) = {-~ ~: : : ~ ~ i }.
0 if x tJ_ A. D- The author is indebted to an anonymous reviewer of an earlier manuscript for pointing
out this example.