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