Diagonals of separately absolutely continuous mappings and their analogues

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 $(g_n{)}_{n=1}^{\mathrm{\infty }}$ of continuous mappings $g_n:X\to Z$ with $\underset{n\to \mathrm{\infty }}{\lim }{g}_n(x)=g(x)$ and \mbox{${\sum }_{n=1}^{\mathrm{\infty }}\mathrm{\parallel }{g}_{n+1}(x)-g_n(x)\mathrm{\parallel }<\mathrm{\infty }$} for every $x\in X$.