On Baire Classification of Strongly Separately Continuous Functions

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 $\sigma(a)=\{x\in X:|\{n\in\mathbb{N}: x_n\ne a_n\}|\lt\aleph_0\}$ 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 $G$.