Pointwise limit of continuous functions is 1) measurable and 2) pointwise discontinuous

Since continuous functions are measurable and pointwise limits of measurable functions are measurable (most measure theory textbooks prove this, see Theorem 4.9 on page 166 of Real analysis by Bruckner, Bruckner & Thomson), Baire class 1 functions are measurable.

On page 20 of the aforementioned book it is proven that every Baire 1 function is continuous except at the points of a set of the first category.

However the converse does not hold: there is a function that is continuous except at the points of a set of the first category but is not in the Baire 1 class. One such function is the characteristic function of the set of the non-endpoints of the Cantor set.

The correct characterization of the Baire 1 class is: A function is Baire 1 if and only if every restriction of the function to any nonempty perfect set has a point of continuity.