Diagonals of separately absolutely continuous mappings and their analogues

TL;DR

This paper characterizes diagonals of separately absolutely continuous functions from an interval to a normed space as limits of sequences of continuous functions with summable differences, extending understanding of such mappings.

Contribution

It provides a precise description of diagonals of separately absolutely continuous mappings in terms of convergent sequences of continuous functions with summable differences.

Findings

Diagonals are exactly limits of sequences of continuous functions with summable differences.

Characterization applies to mappings from an interval to a normed space.

Extends the theory of absolute continuity to multivariable mappings.

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 $\lim\limits_{n\to\infty}g_n(x)=g(x)$ and \mbox{$\sum\limits_{n=1}^{\infty}\|g_{n+1}(x)-g_n(x)\|<\infty$} for every $x\in X$.