The concept of bounded variation has been generalized in many ways. In the frame of functions taking values in Banach space, the concept of bounded semivariation is an important generalization. The aim of this paper is to provide an accessible summary of this notion, to illustrate it with an appropriate body of examples, and to outline its connection with the integration theory due to Kurzweil.

We proved in an earlier paper that the support of the multifractal spectrum of a homogeneously multifractal (HM) measure within $[0,1]$ must be an interval. In this paper we construct a homogeneously multifractal measure with spectrum supported by $[0,1] \cup \{ 2\}$. This shows that there can be a different behaviour for exponents exceeding one.

We also provide details of the construction of a strictly monotone increasing monohölder (and hence HM) function which has exact Hölder exponent one at each point. This function was also used in our paper about measures and functions with prescribed homogeneous multifractal spectrum.

We explore properties of $L^{r}$-derivates with respect to a monotone increasing Lipschitz function. We then define $L^{r}$-ex-major and $L^{r}$-ex-minor functions with respect to a monotone increasing Lipschitz function and use these to define a Perron-Stieltjes-type integral which extends the integral of L. Gordon.

In this note we describe closed subsets of the real line $P\subset {\mathbb R}$ for which there exists a continuous function from $P$ onto $P^2$, called a Peano function. Our characterization of those sets is based on the number of connected components of $P$. We also include a few remarks on compact subsets of $\mathbb{R}^2$ admitting Peano functions, expressed in terms of connectedness and local connectedness.

We study a simple notion of derivative with respect to a function which we assume to be nondecreasing and continuous from the left everywhere. Derivatives of this type were already considered by Young in 1917 and Daniell in 1918, in connection with the fundamental theorem of calculus for Stieltjes integrals. We show that our definition contains as a particular case the delta derivative in time scales, thus providing a new unification of the continuous and the discrete calculus. Moreover, we can consider differential equations in the new sense, and we show that not only dynamic equations on time scales, but also ordinary differential equations with impulses at fixed times are particular cases. We study almost everywhere differentiation of monotone functions and the fundamental theorems of calculus which connect our new derivative with Lebesgue-Stieltjes and Kurzweil-Stieltjes integrals. These fundamental theorems are the key for reducing differential equations with the new derivative to generalized integral equations, for which many theoretical results are already available thanks to Kurzweil, Schwabik and their followers.

We consider two conditions that weaken the closed graph condition and we study their properties. We show that if $X$ is a locally connected Baire space, $Y$ is a separable metrizable space and $f:X \to Y$ is a $w^*$-quasi-continuous, almost continuous and weakly Darboux function, then $f$ is continuous.

We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb{R}$ which is not Baire measurable. We show that if $X$ is a product of normed spaces $X_n$, $a\in X$ and $\sigma(a)=\{x\in X:|\{n\in\mathbb{N}: x_n\ne a_n\}|\lt\aleph_0\}$ is a subspace of $X$ equipped with the Tychonoff topology, then for any open set $G\subseteq \sigma(a)$, there is a strongly separately continuous function $f:\sigma(a)\to \mathbb{R}$ such that the discontinuity point set of $f$ is equal to $G$.

In this paper, we study the behavior of best $L^p$-approximations by algebraic polynomial pairs on unions of intervals when the measure of those intervals tends to zero.

The Fourier dimension is not, in general, stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.

We classify all generalized $\mathcal{A}$-differences of any order $n\geq0$ for which $\mathcal{A}$-continuity at $x$ implies ordinary continuity at $x$. We show that the only $\mathcal{A}$-continuities that are equivalent to ordinary continuity at $x$ correspond to the limits of the form \[ \lim_{h\rightarrow0}A\left[ f(x+rh) +f\left( x-rh\right) -2f(x)\right] +B\left[ f\left( x+sh\right) -f\left( x-sh\right) \right] , \] with $ABrs\neq0$. All other $\mathcal{A}$-continuities truly generalize ordinary continuity.

We study a generalization of Morán’s sum sets, obtaining information about the $h$-Hausdorff and $h$-packing measures of these sets and certain of their subsets.

We introduce a special maximal operator associated to a special variant of Muckenhoupt’s weights. By using this special maximal operator, we can construct the special weights. We also prove a weighted weak-type estimate of the special maximal operator.

We give a simple example of two countable sets $X$ and $Y$ of real numbers such that their upper box-counting dimension satisfies the strict inequality $\dim_B(X\times Y)\lt\dim_B(X)+\dim_B(Y)$.

We prove that there exists a nowhere weakly symmetric function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is everywhere weakly symmetrically continuous and everywhere weakly continuous. Existence of a nowhere weakly symmetrically continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is everywhere weakly symmetric remains open.

A series $\sum x_k$ is $\mathcal{F}$-convergent to $s$ if the sequence $(\sum_{k=1}^n x_k)$ of its partial sums is $\mathcal{F}$-convergent to $s$. We describe filters $\mathcal{F}$ for which $\mathcal{F}$-convergence of a series $\sum x_k$ implies $\mathcal{F}$-convergence to $0$ of the series terms $x_k$. If $(x_k)$ is small enough with respect to a given filter $\mathcal{F}$, then there is an $\mathcal{F}$-subseries $\sum_{k\in I} x_k$ which is absolutely convergent in the usual sense. Filters corresponding to summable ideals, Erdős-Ulam ideals, matrix summability ideals, lacunary ideals and Louveau-Veličković ideals are considered.