Upper semi-continuous functions on the reals are Baire 1, which is readily observed by considering
$$ f_{n}(x):= \sup_{y\in [0,1]}(f(y)- n |x-y| ) \qquad (A).$$
Indeed $f_n$ as in (A) is continuous for each natural $n$ and the pointwise limit of $(f_n)_{n\in \mathbb{N}}$ is $f$.
I am interested to know if 'explicit' representations like (A) exist for larger function classes, or even Baire 1 itself. In particular, for a given Baire 1 function $f$, is there a formula like (A) defining a sequence $(f_n)_{n\in \mathbb{N}}$ with pointwise limit $f$? This purported formula defining $f_n$ may of course involve constructs coming from equivalent definitions of Baire 1 (as in e.g. Baire's characterisation theorem).
