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...


This is impossible. Baire proved that if a function defined on $\mathbb R$ is of Baire class 1, then it is continuous everywhere except, possibly, for a meagre set. And by another Baire's theorem a complement of a meagre set in $\mathbb R$ is dense.

An elementary exposition of this and related results can be found in the nice little book by Oxtoby.

Edit. Another good reference which covers Baire's theorems and provides some historical background is "The calculus gallery" by W. Dunham.

  • $\begingroup$ Could you give a reference? $\endgroup$ – Pete L. Clark Jul 15 '10 at 16:52
  • $\begingroup$ Thanks very much. So I must have seen this before and remembered it but then gotten a little nervous because the results of my google search did not quite match up with my memory. MO to the rescue, as usual. $\endgroup$ – Pete L. Clark Jul 15 '10 at 16:58
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    $\begingroup$ You're welcome. It's a pity that these basic results are not mentioned in the related Wikipedia articles. $\endgroup$ – Andrey Rekalo Jul 15 '10 at 17:43
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    $\begingroup$ When I wrote a paper, First-Class Functions, for the Monthly nearly 20 years ago, I found the book, van Rooij and Schikhof, A Second Course on Real Functions, to be very useful. $\endgroup$ – Gerry Myerson Jul 16 '10 at 0:27
  • $\begingroup$ @Gerry: Thank you for the reference. $\endgroup$ – Andrey Rekalo Jul 16 '10 at 11:15

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