Representations of functions by derivatives

Abstract:

Let $\Delta ’$ be the class of all derivatives. The main goal of this paper is the investigation of the vector space generated by $\Delta ’$ and O’Malley’s class $B_1^ \ast$; this space is identical with our system $[\Delta ’]$. We show, in particular, that each approximately continuous function and each approximate derivative belongs to $[\Delta ’]$ and that $[\Delta ’]$ is the system of all functions of the form $g’ + hk’$, where $g$, $h$ and $k$ are differentiable.