Let $[a,b]$ be an interval in $ℝ$ and let $F$ be a real valued function defined at the endpoints of $[a,b]$ and with a certain number of discontinuities within $[a,b]$. Assuming $F$ to be differentiable on a set $[a,b]\setminus E$ to the derivative $f$, where $E$ is a subset of $[a,b]$ at whose points $F$ can take values $\pm \infty$ or not be defined at all, we adopt the convention that $F$ and $f$ are equal to $0$ at all points of $E$ and show that $\mathrm{𝒦\mathscr{H}}\text{-vt}{\int }_a^{b}f=F\left(b\right)-F\left(a\right)$, where $\mathrm{𝒦\mathscr{H}}\text{-vt}$ denotes the total value of the integral. The paper ends with a few examples that illustrate the...

We present a method of integration along the lines of the Henstock-Kurzweil integral. All $L^r$-derivatives are integrable in this method.

If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\parallel f\parallel ={sup}_I\left|{\int }_{I}f\right|$ where the supremum is taken over all intervals $I\subset ℝ$. Define the translation ${\tau }_x$ by ${\tau }_{x}f\left(y\right)=f(y-x)$. Then $\parallel {\tau }_{x}f-f\parallel$ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\parallel {\tau }_{x}f-f\parallel$ can tend to 0 arbitrarily slowly. In general, $\parallel {\tau }_{x}f-f\parallel \ge \text{osc}f\left|x\right|$ as $x\to 0$, where $\text{osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\parallel {\tau }_{x}F-F\parallel \le \parallel f\parallel \left|x\right|$. An example shows that the function $y\mapsto {\tau }_{x}F\left(y\right)-F\left(y\right)$ need not be in $L^1$. However, if...

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho$-integral, introduced by Jarník and Kurzweil. Let $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ be the space of all strongly $\rho$-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\parallel ·\parallel$. We show that each element in the dual space of $({𝒮}_{\rho }\left(E\right),\parallel ·\parallel )$ can be represented as a strong $\rho$-integral. Consequently, we prove that $fg$ is strongly $\rho$-integrable on $E$ for each strongly $\rho$-integrable function $f$ if and only if $g$ is...