Quasicontinuous and separately continuous functions with values in Maslyuchenko spaces

We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish stability properties of the classes of Maslyuchenko spaces and study the relation of these classes to known classes of generalized metric spaces (such as Piotrowski or Stegall spaces). One of our results says that for any $\aleph_0$-space $Z$ and any separately continuous function $f:X\times Y\to Z$ defined on the product of a topological space $X$ and a second-countable space $Y$, the set $D(f)$ of discontinuity points of $f$ has meager projection on $X$.