Construction of separately continuous functions with given restriction

TL;DR

This paper solves the problem of constructing separately continuous functions on product spaces with prescribed restrictions, showing that for any topological space and first Baire class function, a corresponding separately continuous function exists.

Contribution

It demonstrates the existence of separately continuous functions with given restrictions, including for all first Baire class functions on any topological space.

Findings

Existence of separately continuous functions matching given restrictions.

Construction method applicable to any topological space.

Extension to all first Baire class functions.

Abstract

It is solved the problem on constructed of separately continuous functions on product of two topological spaces with given restriction. In particular, it is shown that for every topological space $X$ and first Baire class function $g:X\to \bf R$ there exists separately continuous function $f:X\times X \to \bf R$ such that $f(x,x)=g(x)$ for every $x\in X$.