**Definition.** A function $f:X\to Y$ between topological spaces is called

$\bullet$ *$G_\delta$-measurable* if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$;

$\bullet$ *$\sigma$-continuous* if $X$ has a countable cover $\mathcal C$ such that $f|C$ is continuous for every $C\in\mathcal C$;

$\bullet$ *piecewise continuous* if $X$ has a countable closed cover $\mathcal C$ such that $f|C$ is continuous for every $C\in\mathcal C$.

By an old result of Jayne and Rogers, each $G_\delta$-measurable maps $f:X\to Y$ between analytic spaces is piecewise continuous and hence $\sigma$-continuous.

**Question.** *Is each $G_\delta$-measurable function between separable metrizable spaces $\sigma$-continuous?*