**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?*



Is each $G_\delta$-measurable map $\sigma$-continuous?
I looked to my own [old paper](https://arxiv.org/pdf/0801.2131) with Bokalo and have found there Example 9.3 answering this problem:


**Example.** Under Martin's Axiom (more precisely, $\mathrm{add}(\mathcal M)=\mathrm{cof}(\mathcal M)$) there exists a bijective function $f:X\to Y$ between zero-dimensional separable metrizable spaces such that $f^{-1}$ is continuous, $f$ is $G\_\delta$-measurable but $f$ is not $\sigma$-continuous.


In fact, the problem was motivated by [this question](https://mathoverflow.net/questions/240215/do-g-delta-measurable-maps-preserve-dimension?rq=1) of Karlova, which is still open.


