This is an idle question motivated by two comments I made to a previous MO question (which I just searched for, unsuccessfully). That question asked if the characteristic function of the rationals is a pointwise limit of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$.

My first reply was that the answer is no: a function is said to be of **Baire class one** if it is a pointwise limit of continuous functions, and (so I remembered) every Baire class one function has a dense set of points of continuity, whereas of course $\bf{1}_{\mathbb{Q}}$ is discontinuous everywhere.

Then I checked up on this result (I am not an analyst, if that is not already clear) and realized that I had misremembered it slightly: the **Baire Characterization Theorem** says that a real function is of Baire class one iff for every nonempty closed subset $S \subset \mathbb{R}$, the restriction of $f$ to $S$ has at least one point of continuity. (Note that in my comment I required that $S$ be perfect, i.e., without isolated points, because some sources state it this way. This hypothesis is however superfluous because an isolated point of a topological space is a point of continuity for every real-valued function on that space.) Taking $S = [a,b]$, we find that either (i) our BC1 function restricted to $S$ is continuous at $c \in (a,b)$ -- meaning that $f$ itself is continuous at $c$, since $S$ is a neighborhood of $c$ -- or (ii) $f|_S$ is continuous at $a$ -- meaning that $f$ itself is *right*-continuous at $a$, or finally (iii) $f|_S$ is continuous at $b$ -- so $f$ itself is *left*-continuous at $b$. So we find that any BC1 function has a dense set of points at which it is either left-continuous or right-continuous (or both). This is good enough to answer the question because $\bf{1}_{\mathbb{Q}}$ has no points of left- or right-continuity.

But later I wondered whether my original claim was actually incorrect:

Do there exist BC1 functions for which the set of points of continuity is not dense? Or even empty?

I did of course try looking on the internet, but I didn't find a reference which gave a solid treatment of Baire classes of real functions: could someone supply one? Moreover, I do see that by taking intervals of the form $(-\infty,b]$ and $[a,\infty)$ a BC1 function must have infinitely many points at which it is left-continuous and infinitely many points at which it is right-continuous. And then I got stumped. Probably this is in the category of things that were much better understood almost one hundred years ago than they are now...