Diagonals of separately continuous functions and their analogs

TL;DR

This paper establishes conditions under which diagonals of separately continuous functions are exactly Baire-one functions and characterizes diagonals of functions with specific continuity and Lipschitz properties.

Contribution

It proves the existence of separately continuous functions with a given Baire-one diagonal and characterizes diagonals of functions with mixed continuity and Lipschitz conditions.

Findings

Diagonals of separately continuous functions are exactly Baire-one functions under certain conditions.

Diagonals of functions continuous in the first variable and Lipschitz in the second are exactly stable first Baire class functions.

The paper extends classical results on function diagonals to more general topological and metric spaces.

Abstract

We prove that for a topological space $X$, an equiconnected space $Z$ and a Baire-one mapping $g:X\to Z$ there exists a separately continuous mapping $f:X^2\to Z$ with the diagonal $g$, i.e. $g(x)=f(x,x)$ for every $x\in X$. Under a mild assumptions on $X$ and $Z$ we obtain that diagonals of separately continuous mappings $f:X^2\to Z$ are exactly Baire-one functions, and diagonals of mappings $f:X^2\to Z$ which are continuous on the first variable and Lipschitz (differentiable) on the second one, are exactly the functions of stable first Baire class.