For any $\alpha ,\beta >0$ with $\alpha +\beta <1$ we provide a simple construction of an $\alpha$-Hölde function $f:[0,1]\to ℝ$ and a $\beta$-Hölder function $g:[0,1]\to ℝ$ such that the integral ${\int }_0^{1}f\mathrm{d}g$ fails to exist even in the Kurzweil-Stieltjes sense.

A fixed point theorem in ordered spaces and a recently proved monotone convergence theorem are applied to derive existence and comparison results for solutions of a functional integral equation of Volterra type and a functional impulsive Cauchy problem in an ordered Banach space. A novel feature is that equations contain locally Henstock-Kurzweil integrable functions.

When a real-valued function of one variable is approximated by its $n$th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{\left(n\right)}$ or $f^{(n+1)}$ are Henstock-Kurzweil integrable. When the only assumption is that $f^{\left(n\right)}$ is Henstock-Kurzweil integrable then a modified form of the $n$th degree Taylor polynomial is used. When the only assumption is that $f^{\left(n\right)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.