On Baire classification of strongly separately continuous functions

TL;DR

This paper explores the Baire classification of strongly separately continuous functions on product spaces, demonstrating the existence of such functions that are not Baire measurable and characterizing their discontinuity sets.

Contribution

It proves the existence of strongly separately continuous functions that are not Baire measurable on countable products of real lines and characterizes their discontinuity sets on certain subspaces.

Findings

Existence of non-Baire measurable strongly separately continuous functions on countable products of real lines.

Construction of strongly separately continuous functions with prescribed discontinuity sets.

Characterization of discontinuity sets of such functions on subspaces of product spaces.

Abstract

We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb R$ which is not Baire measurable. We show that if $X$ is a product of normed spaces $X_n$, $a\in X$ and $\sigma(a)=\{x\in X:|\{n\in\mathbb N: x_n\ne a_n\}|<\aleph_0\}$ is a subspace of $X$, equipped with the Tychonoff topology, then for any open set $G\subseteq \sigma(a)$ there is a strongly separately continuous function $f:\sigma(a)\to \mathbb R$ such that the discontinuity point set of $f$ is equal to~$G$.