Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with the set $\mathbb Z$ of integers?

We recall that a function $f:X\to\mathbb R$ is of the *first Baire class* if it is a pointwise limit of a sequence of continuous functions. By a classical theorem of Baire, a function $f:X\to\mathbb R$ on a Polish space $X$ is of the first Baire class if and only if for any non-empty closed subset $A\subset X$ the restriction $f|A$ has a point of continuity.
It is also well-known that each separately continuous function $f:X\times Y\to\mathbb R$ on the product of two Polish spaces is of the first Baire class.

That is why the affirmative answer to the above problem will imply the affirmative answer to the problem Uniform approximation of separately continuous functions on zero-dimensional spaces .