I thought the following result was well-known, but I can't seem to find it in any standard textbook on real analysis or general topology:
Theorem (★). Let $X$ be a Baire topological space and $f\colon X\to\mathbb{R}$ be a (say, lower) semi-continuous function. Then the set of points of continuity of $f$ is a dense $G_\delta$.
I don't want to assume anything on $X$.
When $X$ is actually a complete metric space, then (★) can be proved (and this is, in fact, how Baire originally did it for $X = \mathbb{R}$) by writing $f$ as the pointwise limit of a pointwise nondecreasing sequence of continuous functions, but this fact does not hold unless $X$ is perfectly normal (see, e.g., Engelking, General Topology (1989), exercise 1.7.15(c)).
Moreover, a direct proof of (★) without writing $f$ as a pointwise limit of continuous functions is not only more general but also simpler. I can write it down for completeness of MathOverflow if someone is interested (or if nobody comes up with a nice reference…), but a sketch is that if we let $V_r := \{x\in X : f(x)>r\}$ then every point of discontinuity $x$ of $f$ is in $\operatorname{closure}(V_r) \setminus V_r$ where $r$ is any rational number strictly between $f(x)$ and $\limsup_{y\to x} f(y)$: so the set of discontinuity points is contained in a countable union of nowhere dense sets.
I found this proof in Kenneth Hardy's PhD thesis, Rings of Normal Functions (McGill University, 1968), where it is prop. 1.20. But this is not a very convenient reference to cite. In the published paper version of his thesis, “On Normal Functions”, General Topology and Appl.2 (1972) 157–163, he cites “an easy extension of a result in [Čech, Topological Spaces (1966), p. 385]”: also not great as a reference.
I looked in various places, e.g., Bourbaki, Topologie Générale, IX, but I couldn't find the statement.
So, does anyone have a reference for (★)?
