I am trying to prove (or disprove) the following assertion:

Consider a probability triple $(X,\mathcal{B},\mu)$, $X$ separable Banach space (complete), $\mathcal{B}$ the Borel $\sigma-$algebra and $\mu$ a countably additive probability measure there on. Let $Y$ be a different separable Banach space.

Given a Baire 1 function $f:X\rightarrow Y$, there exists a $\mu-$almost everywhere continuous function $g:X\rightarrow Y$, such that $f(x)=g(x)$, $\mu-$almost everywhere.

I couldn't find a counterexample so far, even in $\mathbb{R}$, since all the major examples of Baire 1 functions that I know, seems to have this property: the characteristic function of the integers, the characteristic function of the Cantor set, etc.

It is interesting that it is true also for some Baire 2 functions. For example the Dirichlet function, which is equal to 0 $\lambda-$almost everywhere (where $\lambda$ is the Lebesgue measure).