On Totalization of the $H_1$-Integral

Abstract

Based on the total H$_{1}$-integrability concept, which is established in this paper, we shall try to show that at any point of a compact interval $(a,b]$ in $\mathbb{R}$, at which a point function $F$ has no a discontinuity, $F$ is the total H$_{1}$-indefinite integral of a function $dF_{ex}$ being the limit of $\Delta F_{ex}(I)$, where $I \subseteq [a,b]$, on $[a,b]$, without additional hypotheses on $F$. A residue function of $F$ is introduced. The paper ends with a few of examples that illustrate the theory.