Is it true that a bounded real function $f:[0,1]\to[0,1]$ with only countably many discontinuities has to be of Baire class 1, that is pointwise limit of a sequence of continuous functions? Is there a counter-example?


This would be an easy consequence of a theorem stated here: <https://encyclopediaofmath.org/wiki/Baire_theorem>


The theorem, attributed to Baire in that webpage, states that a function is the pointwise limit of a sequence of continuous functions if and only if its restriction on any perfect set has a point of continuity. However, in the original work by Baire added as a reference I could only find a weaker version of this theorem, which does not imply what I am searching for.  

Many thanks!



Baire class 1 and discontinuities
It sounds like you're looking for Baire's characterization theorem. The first reference Google gives me is [here](https://math.ucsd.edu/_files/undergraduate/honors-program/honors-program-presentations/2012-2013/Siuyung_Fung_Honors_Thesis.pdf).


