In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the $p$-sequence, $\{k^{-p}\}_{k=1}^\infty$. We prove that these Cantor sets are $s$-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries of the so-called random stable zonotopes.

Let $f: F \to X$ be a locally bounded function from a closed set $F \subset \R$ to a normed linear space. Then there exists its extension $f^* : \R \to X$ which is differentiable at all points at which $f$ is differentiable. Moreover, $f^*$ is Lipschitz if $f$ is Lipschitz and, in the case $X = \R$, the extension ``preserves Dini derivatives''. The paper partly extends results proved by V. Jarn\'\i k (1923), G. Petruska and M. Laczkovich (1974) and J. Ma\v r\'\i k (1984).

An $n$-th order symmetric Perron type integral is defined and its properties are studied. An integration by parts formula is proved and applied to solve problems related to summable trigonometric series.

This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain $X$ of the functions, the equalities: $D(X)=S_{lf}(X), S(X)=\dim(\Re )$ hold, where $D(X)$ is a ${\cal D}$-number of $X$, $S(X)$ ($S_{lf}(X)$) denotes the Souslin (lf-Souslin) number of $X$ and $\dim(\Re )$ is a Goldie dimension of an arbitrary prime Darboux ring $\Re$.

In this paper we consider the properties of moduli of smoothness of fractional order. The main result of the paper describes the equivalence of the modulus of smoothness and a function from some class.

There is a list of about 50 properties which characterize continuous maps of the interval with zero topological entropy. Most of them were proved by A.N. Sharkovsky [cf., e.g., Sharkovsky et al., Dynamics of One-Dimensional Mappings, Kluwer 1997]. It is also well known that only a few of these properties remain equivalent for continuous maps of the square. Recall, e.g., the famous Kolyada's example of a triangular map of type $2^\infty$ with positive topological entropy. In 1989 Sharkovsky formulated the problem to classify these conditions in a special case of triangular maps of the square. The present paper is a step toward the solution. In particular, we give a classification of 23 conditions in the case of triangular maps of the square which are non-decreasing on the fibers. We show that the weakest is"no homoclinic trajectory", the two strongest, mutually incomparable, are "map restricted to the set of chain recurrent points is not Li and Yorke chaotic" and "every $\omega$-limit set contains a unique minimal set"

We obtain sufficient conditions for the series $\sum\nolimits a_{n}\lambda_n$ to be absolutely summable of order $k$ by a triangular matrix.

The main aim of this paper is to construct an absolutely continuous function with non-$\sigma$-porous graph. This answers the question posed by Foran, whether there exists a function of bounded variation with non-$\sigma$-porous graph. As a consequence we obtain a positive solution of Goffman’s question, whether there exists a function with non-$\sigma$-porous graph such that its Fourier series converges uniformly.

Let $\mathcal{I}$ be an admissible (i.e., proper and containing all finite subsets of $\mathbb{N}$) ideal of subsets of the set $\mathbb{N}^+$ of positive integers. The concept of $\mathcal{I}$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\mathcal{I}$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a boundedly compact metric space which ensures the connectedness of the set of all statistical cluster points of $x$. If $\Gamma_x(\mathcal{I})$ denotes the set of all $\mathcal{I}$-cluster points of the sequence $x$ and $M$ is a set of sequences in a metric space $X$ such that, for each $x\in M$, $\Gamma_x(\mathcal{I})\ne\emptyset$, then the assignment $x\mapsto\Gamma_x(\mathcal{I})$ gives a map $M\to\mathcal{F}$ where $\mathcal{F}$ is the set of all non-empty closed subsets of the space $X$ or $M\to\mathcal{C}$ where $\mathcal{C}$ is a suitable subset of $\mathcal{F}$. In the second part of the paper we study the continuity of this map with respect to the sup-metric on $M$ and some standard hypertopologies on $\mathcal{C}$ (the Vietoris topology, the Fell topology, the proximal topology and the topology given by the Hausdorff metric). We obtain some positive results in the case of locally compact and, particularly, boundedly compact metric spaces.

A function $f:\mathbb R \to \mathbb R$ has the property ${\mathcal M}_1$ (${\mathcal M}_2$) if the restricted function $f | D(f)$ ($f | D_{ap}(f)$) is monotone. ($D(f)$ $D_{ap}(f)$ denotes the set of all discontinuity points [the set of all approximate discontinuity points] of $f$.) In this article I investigate the uniform, pointwise and transfinite limits of sequences of functions with the property ${\cal M}_i$, $i = 1,2$.

Measure theoretic Dirichlet forms on compact subsets of the real line are introduced. Using the technique of Dirichlet--Neumann--bracketing, estimates of the eigenvalue counting functions of the associated measure geometric Laplacians are obtained.

We show that the Continuum Hypothesis implies that for every $ 0 < d_1 \leq d_2 < n $ the measure spaces $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_1}},\mathcal {H}^{d_1})$ and $(\mathbb{R}^n,\mathcal{M}_{\mathcal{H}^{d_2}},\mathcal{H}^{d_2})$ are isomorphic, where $\mathcal{H}^d$ is $d$-dimensional Hausdorff measure and $\mathcal{M}_{d}$ is the $\sigma$-algebra of measurable sets with respect to $\mathcal{H}^d$. This is motivated by the well-known question (circulated by D. Preiss) whether such an isomorphism exists if we replace measurable sets by Borel sets. We also investigate the related question whether every continuous function (or the typical continuous function) is Hölder continuous (or is of bounded variation) on a set of positive Hausdorff dimension.

In 2001, Cs\" ornyei, O'Neil and Preiss proved that the composition of any two Darboux Baire-1 functions $[0,1]\rightarrow [0,1]$ possesses a fixed point, solving a long-standing open problem. In 2004 Szuca proved that this result can be generalized to any $f$ in the class $\cal J$ of functions $[0,1]\rightarrow [0,1]$ with connected $G_\delta$ graph. As a consequence, he proved that for such functions the Sharkovsky theorem is satisfied. As the main result of this paper we prove that as for continuous maps of the interval, any $f$ in $\cal J$ has positive topological entropy if and only if it has a periodic point of period different from $2^n$, for any $n\in\mathbb N$. To do this we show that using Bowen's approach it is possible to define topological entropy for discontinuous maps of a compact metric space with almost all of the standard properties. In particular, the variational principle is true, and consequently, topological entropy is supported by the set of recurrent points. We also develop theory of recurrent, $\omega$-limit, and nonwandering points of functions in $\cal J$ since, in general, standard results from the topological dynamics, are not true. For example, there is a Darboux Baire-1 function $f$ (hence, $f\in\cal J$) such that neither the set of recurrent points nor the set of $\omega$-limit points of $f$ are invariant.

A concept of a general derivative and a notion of bounded variation have been produced leading to the presentation of the Fundamental Theorem of Calculus for the $GH_k$ integral with a characterization of the integral.

The main aim of this paper is to construct an absolutely continuous function with non-s-porous graph. This answers the question posed by Foran, whether there exists a function of bounded variation with Goffman's question, whether there exists a function with non-$\sigma$-porous graph such that its Fourier series converges uniformly.

This paper has two aims. On the one hand, we generalize the notion of sliced measures by means of transversal mappings and study dimensional properties of these measures. On the other hand, as an application of these results, we explain in what sense typical visible parts of a set with large Hausdorff dimension are smaller than the set itself. This is achieved by establishing a connection between dimensional properties of generalized slices and those of visible parts.

Some classes of strongly quasicontinuous functions are investigated.

Green's Theorem is proved using only the geometric (or physical) definition of curl, without the use of partial derivatives. The curl free (conservative) case can then be used to prove Cauchy's Theorem.

We obtain an estimate for the typical Hausdorff dimension of the intersection of a set $E$ with homothetic copies of a set $F$, where $E$ and $F$ are Borel subsets of $\mathbb{R}^{n}$. We apply this to the `distance set problem' for a polyhedral norm on $\mathbb{R}^{n}$, by showing that there are subsets of full dimension with distance set of Lebesgue measure 0.

We study quasi-monotonic functions of the Zygmund-Bary-Stechkin class $\Phi$ with the main emphasis on properties of the index numbers of functions in this class. A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in $\Phi$ known in terms of these indices, are exact in a certain sense. We also single out some special family of none-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0< \alpha\leq \beta <1$, we explicitly construct examples of functions in $\Phi$ for which $\alpha$ and $\beta$ serve as the index numbers. The investigation of properties of non-equilibrated functions in $\Phi$ was evoked by applications of these properties in problems of the normal solvability of some singular integral operators in the spaces with prescribed modulus of continuity.

There is a set $E$ of positive Lebesgue measure and a function nowhere differentiable on $E$ which is differentiable in the $L^p$ sense for every positive $p$ at each point of $E$. For every $p\in(0,\infty]$ and every positive integer $k$ there is a set $E=E(k,p)$ of positive measure and a function which for every $q< p$ has $k$ $L^q$ Peano derivatives at every point of $E$ despite not having an $L^p$ $k$th derivative at any point of $E$.

For a given sequence of measures $\mu_n$ on the circle $\mathbb{T}$ weakly convergent to the Dirac measure, we ask, is it possible to extract a subsequence $n(j)$ such that for any $f$ in the space $L^1 (L^2 ,L^{\infty })$ the convolutions $f\ast\mu_{n(j)}$ converge to $f$ almost everywhere. We show that it is crucial whether the measures are absolutely continuous, discrete or singular (non-atomic).

A function $ f: \mathbb{R}^n \to \mathbb{R}$ satisfies condition $(Q_i(x))$ (resp. $(Q_s(x))$, [$Q_o(x)$]) at a point $x$ if for each real $r > 0$ and for each set $U \ni x$ belonging to the Euclidean topology in $\mathbb{R} ^n$ (resp.~to the strong density topology [to the ordinary density topology]) there is an open set $I$ such that $I \cap U \neq \emptyset $, $f$ is Lebesgue integrable on $I\cap U$ and $$\left|\frac{1}{\mu (U\cap I)}\int_{U \cap I} f(t)dt - f(x) \right| < r.$$ These notions are modifications of quasicontinuity or approximate quasicontinuity. In this article we investigate the limits of sequences of such functions.

It is shown that if $f$ is Henstock-Kurzweil integrable on a compact interval $E$ in ${\mathbb R}^m$, then $f$ is upper semicontinuously integrable on $E$ if and only if there exists an increasing sequence $\{X_n\}$ of closed sets whose union is $E$, and $f |_{X_n}$ is bounded for each positive integer $n$.

We define two interval functions $U_\delta$ and $V_\delta$ using Riemann sums of Henstock integrable functions, as major and minor functions. Then we formulate two dominated convergence theorems for the Henstock integral in the $n$-dimensional space.

We find an analogue of Rolle's Theorem in (real variable) calculus for continuous complex valued functions defined on convex subsets of the complex plane.

We give new tests for series whose terms are defined by iteration of a positive map. Several known results are generalized and unified.

It is shown that each function which can be written as the maximum of two Darboux functions can be written as the maximum of two almost continuous functions as well.

We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are:

1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively.

2. Using Scheepers' notation for selection principles: $\mathsf{S}_fin(\Omega,\Omega^{gp})\cap\mathsf{S}_1(\mathcal{O},\mathcal{O})=\mathsf{S}_1(\Omega,\Omega^{gp})$, and Borel's Conjecture for $\mathsf{S}_1(\Omega,\Omega)$ (or just $\mathsf{S}_1(\Omega,\Omega^{gp})$) implies Borel's Conjecture.

These results extend results of Scheepers and Miller, respectively.

In article [4] D. N. Sarkhel investigates the right preponderant limit of a function and he proves that a such finite limit is of Baire $1$ class. In this article I generalize this Sarkhel's result.

We construct a $C^1$-function $f\colon [0,1]\to \mathbb{R}$ such that for almost all $x\in(0,1)$, there is $r>0$ for which $f(y)>f(x)+f'(x)(y-x)$ when $y\in(x,x+r)$ and $f(y)< f(x)+f'(x)(y-x)$ when $y\in(x-r,x)$. The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on $\mathbb{R}^n$. We also discuss the tangential behavior of typical $C^1$-functions, using an improvement of Jarník's theorem on essential derived numbers

We construct a sequence of continuous functions $(h_n)$ on any given uncountable Polish space, such that $\sum h_n$ is divergent everywhere, but for any sign sequence $(\epsilon_n) \in \{-1, +1\}^\mathbb{N}$ which contains infinitely many $-1$ and $+1$ the series $\sum \epsilon_n h_n$ is convergent at at least one point. We can even have $h_n \to 0$, and if we take our given Polish space to be any uncountable closed subset of $\mathbb{R}$, we can require that every $h_n$ be a polynomial. This strengthens a construction of Tamás Keleti and Tamás Mátrai.

In any complete separable metric space, the Boolean algebra $(s)/(s_0)$ of Marczewski sets modulo the Marczewski null sets is complete. Using this fact, we show that a simple construction solves two known problems in real analysis: the existence of an $(s_0)$-set which is not Lebesgue measurable and does not have the Baire property, and a function which is not $(s)$-measurable, but whose graph is an $(s_0)$-set. Using a similar idea, we also present a short proof that the Boolean algebra of universally measurable sets modulo the sets universally of measure zero is not complete.

A function $h : \mathbb{R}^n \to \mathbb{R}^k$ is called a Hamel function if it is a Hamel basis for $\mathbb{R}^{n+k}$. We prove that there exists a Hamel function which is finitely continuous (its graph can be covered by finitely many partial continuous functions). This answers the question posted in [3].

In this article we describe the products of two unilaterally continuous regulated functions.

We discuss relations between the Titchmarsh convolution theorem, the Kierat-Skornik theorem on convolutions and the Brodski-Sakhnovich-Donoghue theorem on cyclic vectors of the Volterra integration operator $V$ given by $Vf\left( x\right) =\int_{0}^{x}f\left( t\right) dt,$ on a space $L^{p}[0,1],\;1\leq p<\infty .$