Find a continuous function with a prescribed continuity set

It's known that for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ the set of points of discontinuity must be an $F_{\sigma}$.

In the book "Understanding Analysis" by Abbott is stated in page 128 that this property is "sharp"; that is, for every $D \subseteq \mathbb{R}$ in the class $F_{\sigma}$ there exists a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that possesses $D$ as its set of discontinuity.

The question is the following: in that same page of the book Abbott says this last result has been proved by W.H. Young in 1903 but it gives no more precisions about that.

Does anybody know which paper Abbott refers to? Another reference? I'm interested in the construction of the function.

This old question is related but no references there: Possible subsets of reals that equal the set of continuity of a function

Thanks

• For Young's paper, see reference [51] in my 20 December 2006 sci.math post References for Continuity Sets. This post also gives a lengthy list of references for the topic. Jun 5 '13 at 14:00
• Thanks, Dave. Very good post there. Found useful the exposition in Hobson's "The Theory of Functions of a Real Variable and The Theory of Fourier's Series"
– PIP
Jun 6 '13 at 20:18
• Of course 'continuous' very definitely doesn't belong in the title. Sep 20 '19 at 2:07
• The link I gave above for my 20 December 2006 sci.math post no longer works, but this google-groups link works, at least at the time of my present comment. Sep 19 '20 at 7:48

1 Answer

Probably W. H. Young, Über die Einteilung der unstetigen Funtionen und die Verteilung ihrer Stetigkeitspunkte, S.-B. Akad. Wiss. Wien Math.-Natur. K. Abt. IIA 112 (1907), 1307- 1311. See Richard Bolstein, Sets of points of discontinuity, Proc Amer Math Soc 38 (1973) 193-197, which should be available on the AMS website.