Using AC, one easily defines a function $F:\mathbb R\to \mathbb R$ such that the $F$-image of any real interval $(a,b)$ ($a<b$) is equal to $\mathbb R$. (Equivalently, the $F$-preimage of any real singleton has to be a dense set in $\mathbb R$.) Does there exist a Borel-measurable $F$ with this property?
