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Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\mathbb R\to\mathbb R$ that is continuous exactly on $S$.

I'm trying to find a published reference for this result. Note: the wikipedia page on Thomae's function mentions an equivalent result, phrased in terms of $F_\sigma$'s, without giving a reference. So it's clear the result is well-known; but I'd like a reference in an article or book (even as an exercise) rather than just the wikipedia page.

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    $\begingroup$ Have you tried Kechris' book? Specifically Theorem 3.8 seems relevant here. $\endgroup$ Commented Oct 28, 2014 at 18:55
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    $\begingroup$ Are you referring to Classical Descriptive Set Theory? Theorem 3.8 is Kuratowski's theorem, and I don't see how it easily implies the result I quoted. (I'm not looking for a proof of the result, but for a source where it appears, preferably in a form that's not too distant from what I quoted.) $\endgroup$ Commented Oct 28, 2014 at 19:17
  • $\begingroup$ Well, take $A$ to be the $G_\delta$ set and take its characteristic function. $\endgroup$ Commented Oct 28, 2014 at 19:26
  • $\begingroup$ Sorry if I'm being dense but the theorem then returns a set $G$ possibly bigger than the starting set, and a function $G\to \mathbb R$ that is continuous on all of $G$. $\endgroup$ Commented Oct 28, 2014 at 19:34
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    $\begingroup$ A nice argument can be found in A second course on real functions by van Rooij and Schikhof: First, prove that if $O$ is open, there is a function $f_O$ that is zero outside of $O$, and discontinuous precisely at points in $O$. Second, prove that if $F$ is an $F_\sigma$ that contains no intervals, say $F=\bigcup_n C_n$, each $C_n$ closed nowhere dense, then $f_F=\sum_n \chi_{C_n}/2^n$ is continuous precisely at points not in $F$. Third, any $F_\sigma$ set is the disjoint union of an $F_\sigma$ set $F$ that contains no intervals, and an open set $O$, and take $f=f_O+f_F$. $\endgroup$ Commented Oct 29, 2014 at 5:53

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The result you are looking for can be found in most advanced references in topology or analysis. Since I am away from my office at the time of this writing, I will provide you with a source readily available on the internet: Theorem 7.2 (p.30) in Oxtoby's Measure and Category.

Note that Theorem 7.2 shows that any $F_{\sigma}$ set is precisely the set of discontinuities of some real valued function on $\Bbb{R}$; the result you want the reference for is equivalent to Theorem 7.2 thanks to De Morgan Laws. This theorem, I suspect, is due to Lebesgue, but at the moment I do not have a reference at hand to corroborate my suspicion.

Addendum: According to this source (see history of Theorem 2') the above result was first proved by William H. Young in 1903, and perhaps independently by Lebesgue in 1904.

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