A characterization of sets of discontinuity points of separately continuous functions of many variables on products of metrizable spaces

TL;DR

This paper characterizes the sets of discontinuity points for separately continuous functions on products of metrizable spaces, showing they are unions of specific meager $F_{\sigma}$-sets.

Contribution

It provides a complete characterization of discontinuity sets for separately continuous functions on product spaces of metrizable spaces.

Findings

Discontinuity sets are unions of $F_{\sigma}$-sets that are locally projectively meager.

The characterization is both necessary and sufficient.

The results extend understanding of discontinuity behavior in multivariable functions.

Abstract

It is shown that a set in product of $n$ metrizable spaces is the discontinuity points set of some separately continuous function if and only if this set can be represented as the union of a sequence of $F_{\sigma}$-sets which are locally projectively meager.