On a problem of Talagrand concerning separately continuous functions

TL;DR

This paper constructs a specific separately continuous function on a product space that demonstrates a negative answer to Talagrand's problem, showing such functions can lack continuity on large subsets.

Contribution

It provides a counterexample to Talagrand's problem by constructing a separately continuous function with no continuous restriction on any non-meager Borel subset.

Findings

Constructed a separately continuous function with no continuous restriction on non-meager Borel sets.

Demonstrated a negative solution to Talagrand's problem.

Showed limitations of separate continuity in product spaces.

Abstract

We construct a separately continuous function $e:E\times K\to\{0,1\}$ on the product of a Baire space $E$ and a zero-dimensional compact space $K$ such that no restriction of $e$ to any non-meager Borel set in $E\times K$ is continuous. The function $e$ provides a negative solution of Talagrand's problem in \cite{T}.