In this paper we introduce some notions which are related to quasi-continuity and a strong Świątkowski property. We examine the basic properties, the uniform closure, and several varieties of maximal classes for the family of internally strong Świątkowski functions. Moreover we study when the function \(f\) can be expressed as the sum of two internally quasi-continuous functions and the sum of two internally strong Świątkowski functions.

In the paper we prove that some natural modifications (for instance, images under some functions, Cartesian products, quotient sets) of certain types of \(\mu\)-shadings (or shadings), are other examples of \(\mu\)-shadings (or shadings). The studies of shadings and \(\mu\)-shadings were initiated by R. Mabry in 1990. Our work is a continuation of his and K. Neu's research in this field. In particular, we solve one problem posed by R. Mabry.

Let \(X\) be a compact topological space and \(T:X\rightarrow X\) a continuous map. Y.N. Dowker, F.G. Friedlander and A.N. Sharkovsky, independently, introduced and studied the notion of \emph{\(T\)-connectedness}. In particular, they showed that any \(\omega\)-limit set of a dynamical system \((X,T)\) is \(T\)-connected. Let \(\big(C(X),F\big)\) be the induced dynamical system of a given system \((X,T)\), where \(C(X)\) is the hyperspace of all compact connected subsets of \(X\) and \(F:C(X)\rightarrow C(X)\) is the induced map of \(T\). In this paper we give a characterization of the induced-map-connected subsets of \(C(I)\), where \(I\) is a compact interval. The characterization is given via the structure of the \(\omega\)-limit sets (located on a fiber) of continuous \emph{triangular induced maps} on \(I\times C(I)\).

Let \(\mu\) be the attracting measure of a condensation system consisting of a finite system of conformal mappings associated with a probability measure \(\gn\) which is the image measure of an ergodic measure with bounded distortion. We have shown that for a given \(r\in (0,+\infty)\) the lower and the upper quantization dimensions of order \(r\) of \(\mu\) are bounded below by the quantization dimension \(D_r(\gn)\) of \(\gn\) and bounded above by a unique number \(\gk_r\in (0, +\infty)\) where \(\gk_r\) has a relationship with the temperature function of the thermodynamic formalism that arises in multifractal analysis of \(\mu\).

We consider the descriptive complexity of some subsets of the infinite permutation group \(S_\infty\) which arise naturally from the classical series rearrangement theorems of Riemann, Levy, and Steinitz. In particular, given some fixed conditionally convergent series of vectors in Euclidean space \(\mathbb{R}^d\), we study the set of permutations which make the series diverge, as well as the set of permutations which make the series diverge properly. We show that both collections are \(\boldsymbol\Sigma^0_3\)-complete in \(S_\infty\), regardless of the particular choice of series.

Dealing with the technically motivated concept of convolution operators in bounded regions of \(\mathbb{R}^{N}\) with an underlying nearby boundary condition we extend a formerly proved result about the existence and uniqueness of suitable solutions for dimension \(N\leq 2\) to arbitrary dimensions \(N\). Thus, a first substantial result in a sufficiently generalized form, beyond the very specific case of rectangular regions, is established in this field. The result can also be seen as a generalized maximum principle for so called \(k\)-harmonic functions where \(k\) is the kernel of the given convolution operator.

The class of linearly continuous functions from \(f \colon \mathbb{R}^n \to \mathbb{R}\), that is, having continuous restrictions \(f \upharpoonright \ell\) to every straight line \(\ell\), have been studied since the dawn of the twentieth century. In this paper we refine a description of the form that the sets \(D(f)\) of points of discontinuities of such functions can have. It has been proved by Slobodnik that \(D(f)\) must be a countable union of isometric copies of the graphs of Lipschitz functions \(h\colon K\to\mathbb{R}\), where \(K\) is a compact nowhere dense subset of \(\mathbb{R}^{n-1}\). Since the class \(\mathcal{D}^n\) of all sets \(D(f)\), with \(f \colon \mathbb{R}^n \to \mathbb{R}\) being linearly continuous, is evidently closed under countable unions as well as under isometric images, the structure of \(\mathcal{D}^n\) will be fully discerned upon deciding precisely which graphs of the Lipschitz functions \(h\colon K\to\mathbb{R}\), \(K\subset\mathbb{R}^{n-1}\) being compact nowhere dense, belong to \(\mathcal{D}^n\). Towards this goal, we prove that \(\mathcal{D}^2\) contains the graph of any such \(h\colon K\to\mathbb{R}\) whenever \(h\) can be extended to a \(C^2\) function \(\bar h\colon\mathbb{R}\to\mathbb{R}\). Moreover, for every \(n\gt 1\), \(\mathcal{D}^n\) contains the graph of any \(h\colon K\to\mathbb{R}\), where \(K\) is closed nowhere dense in \(\mathbb{R}^{n-1}\) and \(h\) is a restriction of a convex function \(\bar h\colon\mathbb{R}^{n-1}\to\mathbb{R}\). In addition, we provide an example, showing that the above mentioned result on \(\mathcal{C}^2\) functions need not hold when \(\bar h\) is just differentiable with bounded derivative (so Lipschitz).

We investigate the multi-fractal analysis of (large) convolution powers of probability measures on \(\mathbb{R}\). If the measure \(\mu \) satisfies \((N)\) supp\(\mu =[0,N]\) for some \(N\), then under weak assumptions there is an isolated point in the multi-fractal spectrum of \(\mu ^{n}\) for sufficiently large \(n\). A formula is found for the limiting behaviour (as \(n\rightarrow \infty \)) of the \(L^{q}\)-spectrum of \(\mu ^{n}\) and this is related to the limit of the energy dimension of \(\mu ^{n}\) when \(q\geq 1\).

We study spaces of homogeneous type, and especially the theory of atoms, on the boundary of a domain in \(\CC^n\). We are particularly interested in atoms for small \(p\), which must satisfy a higher-order moment condition. We have an axiomatic presentation of these ideas which avoids a lot of the usual nasty calculations. Examples show that this new theory is consistent with existing particular instances of atoms.

In this note we prove that 1/2 is an upper bound for the Hausdorff dimension of the intersection of the graph of Takagi's function with any line of integer slope.

We present a short proof of the existence of universal functions for period-doubling and critical golden-mean circle maps for all degrees of criticality \(d \gt 1\). The method is based on H. Epstein's Herglotz-function technique.

This is an expository note, hopefully accessible to students, on how continuous convergence and convergence-theroretic techniques can provide insight on the classical Arzelá-Ascoli theorem.

A new concept of McShane-tightness is introduced to pass from scalar (alias weak) McShane integrability to Pettis integrability. It is used also to derive a scalar McShane version of the Vitali theorem.

In this work, we extend the Hermite-Hadamard inequality to a new class of functions which do not satisfy the convex property. This result will be applied to both Haber and Fejér inequalities.

Some additive properties of the following three families of “pathological” functions are briefly discussed: continuous nowhere differentiable functions, Sierpiński-Zygmund functions, and absolutely nonmeasurable functions.

The paper proves a general theorem that relates some modes of convergence, such as pointwise a.e. convergence, to weak convergence in any space with the weak Banach-Saks property. Some results that follow immediately from the main theorem are given for specific functional spaces.

We show that for any set of reals \(X\) there is a \(Y \subseteq X\) such \(X\) and \(Y\) have same Lebesgue outer measure and the distance between any two distinct points in \(Y\) is irrational.

In this paper we give a sufficient condition for the strongly separately continuous functions to be continuous on \(l^{2}\). Further we shall give notions of a separately quasicontinuous function \(f:l^2\to R\) and its properties. At the end we will expecting to determining sets \(M \subset l^{2}\) for the class of separately continuous functions on \(l^{2}\).