Diagonals of separately absolutely continuous mappings and their analogues

Abstract

We prove that, for an interval $X\subseteq \mathbb{R}$ and a normed space $Z$ diagonals of separately absolute continuous mappings $f:X^2\to Z$ are exactly such mappings \mbox{$g:X\to Z$} that there is a sequence $(g_n{)}_{n=1}^{\infty }$ of continuous mappings $g_n:X\to Z$ with $\underset{n\to \infty }{\lim }{g}_n(x)=g(x)$ and \mbox{$\sum _{n=1}^{\infty }\Vert g_{n+1}(x)-g_n(x)\Vert <\infty$} for every $x\in X$.