  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~$G$.
