In this paper we characterize products of four or more extra strong Świątkowski functions.

For \(A=\left( \begin{array}[pos]{ clrr} 0 & 1 \\ a & 0 \end{array} \right),\) where \(a\) is an integer such that \(|a|>1\) and a natural number \(d\) satisfying \(L=(|a|-1)d,\) we obtain that the product \(W\times Q\) of a measurable set \(W\) of the Lebesgue measure \(2\pi L,\) and a measurable set \(Q\) in \(\mathbb R\) such that \(Q\subset aQ,\) is an MRA \(A\)-multiwavelet set of order \(Ld\) in \(\mathbb R^2\) if and only if \(W\) is an \(a\)-multiwavelet set of order \(L\) and \(Q\) is an \(a\)-multiscaling set of order \(d\) associated with \(W.\)

A criterion of continuity and monotonicity of one-to-one pointwise monotonic functions is proved. We apply them in the theory of generalized subadditive functions. Some open problems are presented.

Abel derivative of order \(k\) is introduced and the first order Abel derivative is studied. Using Abel derivative some monotonicity results are obtained.

We prove that every locally defined operator mapping the space of continuous and bounded \(\varphi \)-variation functions into itself is a Nemytskij composition operator.

We supplement the well known upper and lower box-counting product inequalities to give the new product formula \begin{align*} \dim_{LB}F + \dim_{LB}G &\leq \dim_{LB}\left(F\times G\right) \\ &\leq \min\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ &\leq \max\left(\dim_{LB}F + \dim_{B}G,\dim_{B}F + \dim_{LB}G\right)\\ & \leq \dim_{B}\left(F\times G\right)\\ &\leq \dim_{B}F+\dim_{B}G \end{align*} for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products.

We adapt a construction taken from L. Motto Ros and B. Semmes, A new proof of a theorem of Jayne and Rogers, Real Anal. Exchange 35(1) (2009/2010), 195-204 in order to correct a mistake contained in the first part of the same paper. As a byproduct of the new construction, the Jayne-Rogers theorem is extended to functions whose range is a regular topological space, and a theorem of Solecki which sharpens the Jayne-Rogers theorem for separable metric spaces is extended to the non-separable context.

We prove that for every separately twice differentiable function \(f:\mathbb {R}^2\to\mathbb{R}\) with that \(f''_{xx}=f''_{yy}\) there exist twice differentiable functions \(\varphi, \psi:\mathbb R\to\mathbb{R}\) such that \(f(x,y)=\varphi(x+y) + \psi(x-y)\).

Given a density \(d\) defined on the Borel subsets of \([0,\infty),\) the limit in density of a function \(f:[0,\infty)\rightarrow\mathbb{R}\) is zero (abbreviated, \((d)\)-\(\lim_{x\rightarrow\infty}f(x)=0)\) if there exists a set \(S\) of zero density such that \(f(x)\rightarrow0\) as \(x\ \)runs to \(\infty\) outside \(S\). It is proved that the behavior at infinity of every Lebesgue integrable function \(f\in L^{1}(0,\infty)\) satisfies the relations \[ (d^{(n)})-\lim_{x\rightarrow\infty}\left( \prod\nolimits_{k=0}^{n}\ln ^{(k)}x\right) f(x)=0, \] where \((d^{(n)})_{n}\) is a scale of densities including the usual one, \(d^{(0)}(A)=\lim_{r\rightarrow\infty}\frac{m\left( A\cap\lbrack0,r)\right) }{r}.\)

According to the Vitali-Carathéodory theorem, the integral of a finite summable function \(f\) on a measurable set may be approximated by the integral of a sum of lower and upper semicontinuous functions. In the case, that \(f\) is a Baire one function, we give the answer to the following question: is there a lower semicontinuous function \(l\) and a upper semicontinuous function \(u\) such that \(f=l+u\) almost everywhere? The answer is in general negative.

We construct a variety of mappings from the unit interval \(\mathcal{I}\) into \(\mathcal{L}^p([0,1]),1\leq p<\infty,\) to generalize classical examples of \(\mathcal{L}^p\)-converging sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of \(\mathcal{I}\), we obtain examples which are \(\mathcal{L}^p\)-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving a Lusin-type theorem, namely that if almost every function in the image is continuous, then we can remove a set of arbitrarily small measure from the index set \(\mathcal{I}\) and establish pointwise continuity in the remainder.

For a family \(\{k_\alpha \,\vert \,\alpha \in I\}\) of real \(\mathcal{C}^2\) functions defined on \(U\) (\(I\), \(U\) — open intervals) and satisfying some mild regularity conditions, we prove that the mapping \(I \ni \alpha \mapsto k_\alpha^{-1}\bigl(\sum_{i=1}^n w_i k_\alpha(a_i)\bigr)\) is a continuous bijection between \(I\) and \((\min\underline{a}, \,\max\underline{a})\), for every fixed non-constant sequence \(\underline{a} = \bigl(a_i\bigr)_{i=1}^n\) with values in \(U\) and every set, of the same cardinality, of positive weights \(\underline{w} = \bigl(w_i\bigr)_{i=1}^n\). In such a situation one says that the family of functions \(\{k_\alpha\}\) generates a scale on \(U\). The precise assumptions in our result read (all indicated derivatives are with respect to \(x \in U\))

(i) \(k'_\alpha\) vanishes nowhere in \(U\) for every \(\alpha \in I\),

(ii) \(I \ni \alpha \mapsto \frac{k''_\alpha(x)}{k'_\alpha(x)}\) is increasing, 1-1 on a dense subset of \(U\) and onto the image \(\mathbb{R}\) for every \(x \in U\).

This result makes possible three things: 1) a new and extremely short proof of the classical fact that power means generate a scale on \((0,+\infty)\), 2) a short proof of a fact, which is in a direct relation to two results established by Kolesárová in 2001, that, for every strictly increasing convex and \(\mathcal{C}^2\) function \(k \colon (0,\,1) \to (0,\,+\infty)\), the class \(\{\mathfrak{M}_{k_\alpha}\}_{\alpha \in (0,\,+\infty)}\) of quasi-arithmetic means (see Introduction for the definition) generated by functions \(k_\alpha\), \(k_\alpha(x) = k(x^\alpha)\), \(\alpha \in (0,\,+\infty)\), generates a scale on \((0,1)\) between the geometric mean and maximum (meaning that, for every \(\underline{a}\), \(\underline{w}\), if \(s \in \bigl(\prod_{i = 1}^n a_i^{\,w_i},\,\max(\underline{a})\bigr)\) then there exists exactly one \(\alpha\) such that \(\mathfrak{M}_{k_\alpha}(\underline{a},\underline{w}) = s\)). 3) a brief proof of one of the classical results of the Italian statistics' school from the 1910-20s that the so-called radical means generate a scale on \((0,\, +\infty)\).

The paper is devoted to the study of limiting behaviour of Besov capacities \(\text{cap}(E;B_{p,q}^\alpha),\ (0<\alpha <1)\) of sets in \(\mathbb{R}^n\) as \(\alpha\to 1\) or \(\alpha\to 0.\) Namely, let \(E\subset \mathbb{R}^n\) and \[ J_{p,q}(\alpha, E)=[\alpha(1-\alpha)q]^{p/q}\,\text{cap}(E;B_{p,q}^\alpha). \] It is proved that if \(1\le p<n,\,\,1\le q<\infty,\) and the set \(E\) is open, then \(J_{p,q}(\alpha, E)\) tends to the Sobolev capacity \(\text{cap}(E;W_p^1)\) as \(\alpha\to 1\). This statement fails to hold for compact sets. Further, it is proved that if the set \(E\) is compact and \(1\le p,q<\infty\), then \(J_{p,q}(\alpha, E)\) tends to \(2n^p|E|\) as \(\alpha\to 0\) (\(|E|\) is the measure of \(E\)). For open sets it is not true.

There are a few nonequivalent definitions of preponderant continuity \cite{bru1,den1,gra,kow,oma}. In this paper we investigate uniform limits of preponderantly continuous functions. We show that preponderantly continuous functions in the O’Malley sense are closed under uniform limits, while the family of preponderantly continuous functions in the Denjoy sense are not closed under uniform limits and find closure of this family in the topology of uniform convergence. Finally, we prove that the set of preponderantly continuous in Denjoy sense functions is a first category subset of its uniform closure.

The cover table of contents for Volume 21 Number 1 is missing an entry. It should include the following title which appeared beginning on page 258.

A. Maliszewski, On the Differences of Darboux Upper Semicontinuous Functions

The editors apologize for the omission.