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 $f\in L^1$ then $\parallel {\tau }_x{F-F\parallel }_1\le {\parallel f\parallel }_1\left|x\right|$....

It is shown that a monotone function acting between euclidean spaces $R^n$ and $R^m$ is continuous almost everywhere with respect to the Lebesgue measure on $R^n$.

Let the spaces ${𝐑}^m$ and ${𝐑}^n$ be ordered by cones $P$ and $Q$ respectively, let $A$ be a nonempty subset of ${𝐑}^m$, and let $f:A⟶{𝐑}^n$ be an order-preserving function. Suppose that $P$ is generating in ${𝐑}^m$, and that $Q$ contains no affine line. Then $f$ is locally bounded on the interior of $A$, and continuous almost everywhere with respect to the Lebesgue measure on ${𝐑}^m$. If in addition $P$ is a closed halfspace and if $A$ is connected, then $f$ is continuous if and only if the range $f\left(A\right)$ is connected.

The problem of continuous dependence for inverses of fundamental matrices in the case when uniform convergence is violated is presented here.

We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

In this paper we use a generalized version of absolute continuity defined by J. Kurzweil, J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exch. 17 (1992), 110–139. By applying uniformly this generalized version of absolute continuity to the primitives of the Henstock-Kurzweil-Pettis integrable functions, we obtain controlled convergence theorems for the Henstock-Kurzweil-Pettis integral. First, we present a controlled convergence theorem for...

We introduce an ap-Henstock-Kurzweil type integral with a non-atomic Radon measure and prove the Saks-Henstock type lemma. The monotone convergence theorem, ${\mu }_{\mathrm{ap}}$-Henstock-Kurzweil equi-integrability, and uniformly strong Lusin condition are discussed.