Using fractal sets and Minkowski contents we extend the repertoire of Lebesgue integrable functions to those with large singular sets. A new method of constructing fractal sets is proposed, using a class of absolutely continuous functions, called swarming functions. We obtain bounds on Minkowski contents of fractals in terms of two natural parameters contained in $[-\infty,\infty]$, called the upper and lower dispersions of the fractal. Assuming that upper and lower box dimensions of a fractal are equal, we show that if the difference of dispersions is sufficiently large, then the set is not Minkowski measurable. Fractals with nondegenerate $d$-dimensional Minkowski contents (i.e., contained in $(0,\infty)$) are characterized as those with nondegenerate dispersions (i.e., contained in $(-\infty,\infty)$). The Weierstrass function, a class of affine fractal functions and a class of McMullen's sets have nondegenerate Minkowski contents. Also some classes of spirals of focus and limit cycle type in the plane are shown to be Minkowski measurable. Using swarming functions we can easily construct fractal sets with maximally separated lower and upper box dimensions, and a pair of fractal sets with maximal instability of lower box dimension with respect to union. We also study gauge functions associated with fractals having degenerate Minkowski contents, and obtain new integrability criteria for a class of singular integrals.

In this paper, we establish a summability factor theorem for summability $|A,\delta|_k$ as defined in (\ref{eq2}) where $A$ is a lower triangular matrix with non-negative entries satisfying certain conditions. Our paper is an extension of a result of Bor and Seyhan (1999) using definition (1) below.

The main purpose of the paper is to extend Maximoff's theorem for real functions to the multivalued maps case.

Theorem (Maximoff [5]) Let $f : \mathbb{R}\to\mathbb{R}$ be a first Baire class function. Then $f$ has the Darboux property if and only if $f$ has a perfect road at each point.

The concept of the GAP-integral was introduced by the authors $[5]$. In this paper some convergence theorems for the GAP-integral are presented.

This paper completes the results of the paper [8]. We will investigate mutual relations between classes of functions which are continuous $\Im$-a.e. with respect to various $\sigma$-ideals of subsets of $\mathbb{R}^k$.

Let $\overline{x} \in [0,1)^2$. In this paper we find the rate at which knowledge about the partition elements $\overline{x}$ lies in for one sequence of partitions determines the partition elements it lies in for another sequence of partitions. This rate depends on the entropy of these partitions and the geometry of their shapes, and gives a two-dimensional version of Lochs' theorem.

This paper introduces a notion of equivalence that links closed relations and quasicontinuous functions; we examine classes of quasicontinuous functions that have the same set of continuity points. In doing so, we show that every minimal closed relation is the closure of a quasicontinuous function and vice-versa. We also show that this notion is of use in dynamical systems. Every quasicontinuous function is equivalent to one that is measurable, and under certain circumstances---in fact, under just those circumstances that appear most often in the dynamics literature---it is equivalent to a quasicontinuous function that has an invariant measure.

We characterize differentiability of a map $f:\mathbb{R\rightarrow R}$ in terms of continuity of a canonically associated map $\widehat{f}$. To characterize pointwise differentiability of $f,$ both the domain and range of $\widehat{f}$ can be made topological. However, the global differentiability of $f$ is characterized by the continuity of $\widehat{f}$ whose domain is topological but whose range is a convergence space.

The Ward and Perron integrals, with respect to abstract derivation bases, for Riesz-space-valued functions are introduced. It is shown that they are equivalent to the respectively defined Henstock-Kurzweil integral.

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh-Fourier series is of type $(H^{\#},L_{1})$ provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in $ H^{\#}$, so by the well-known density argument we have that $t_{2^{n},2^{m}}f\left( x^{1},x^{2}\right) \rightarrow f\left( x^{1},x^{2}\right) $ a. e. as $m,n\rightarrow \infty$ for all $f\in H^{\#} (\supset L\log^{+} L$). We also prove the sharpness of this result. Namely, For all measurable function $\delta :[0,+\infty) \to [0,+\infty) , \, \lim_{t\to\infty}\delta(t)=0$ we have a function $f$ such as $f\in \llogld (L)$ and the two-dimensional Nörlund logarithmic means do not converge to $f$ a.e. (in the Pringsheim sense) on $I^2$.

A useful modification of the definition of the Kurzweil integral is presented.

In this article, I propose a new property $(a)$ of functions $f:X \to Y$, where $X$ and $Y$ are metric spaces. A function $f:X \to Y$ has the property $(a)$ if for each real $\eta>0$, the union $\bigcup\limits_{x\in X}(K(x,\eta)\times K(f(x),\eta))$ contains the graph of a continuous function $g:X \to Y$ and $K(x,r)$ denotes the open ball $\{t\in X:\rho_X(t,x)<r\}$ with center $x$ and radius $r>0$. The class of functions with the property $(a)$ contains all functions almost continuous in the sense of Stallings and all functions graph continuous. Moreover, I examine the sums, the products, and the uniform and discrete limits of sequences of functions from this class.

We prove that two natural definitions of residuality of families of $\mathcal{F}_\sigma$ sets are equivalent. We make use of the Banach-Mazur game in the proof.

In this paper we begin the search for appropriate definitions of first-return continuity and first-return approachability for functions of several real variables. We do this by first reviewing the one dimensional theory and then considering several candidate definitions in higher dimensions.

A function $f\colon{\mathbb{R}}\to{\mathbb{R}}$ is said to be vertically rigid provided its graph $G(f)=\{{\langle} x,f(x){\rangle}\colon x\in{ \mathbb{R}}\}$ is isometric with the graph of the function $kf$ for every non-zero $k\in{\mathbb{R}}$. We show that the group homomorphisms $f$ from ${ \langle}{\mathbb{R}},+{\rangle}$ into ${\langle}{\mathbb{R}}^+,\cdot{\rangle} $ is vertically rigid if and only if it is an epimorphism. Some other examples of vertically rigid functions will also be given. A problem of characterizing all vertically rigid functions remains open.

We present a generalization of the following result of Y. Benyamini. There is a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that for each (/$x_n)_{n \in \mathbb{Z}}\in [0,1]^\mathbb{Z}$, there is $t \in \mathbb{R}$ such that $x_n=f(t+n)$ for all $n\in \mathbb{Z}$.

In this paper, we obtain two inequalities for generalized rational functions of one variable in $L^p$ spaces when a partition of the domain with a suitable number of measurable subsets is considered.

A new characterization of subcontinuity of functions is given. Relations among subcontinuity, local boundedness and local compactness are studied. The set of points of subcontinuity of functions is investigated.

In a zero-dimensional Polish space, the Borel sets are generated from the clopen sets by repeatedly applying the operations of countable disjoint union and complementation. Here we look at topologically restrictive versions of the general countable disjoint union of sets, and obtain ``construction principles'' for $\mathbf{\Delta} \begin{smallmatrix} 0 \\ 3 \end{smallmatrix}$, i.e., sets which are both $\mathcal{F}_{\sigma\delta}$ and $\mathcal{G}_{\delta\sigma}$.

Assuming Martin's Axiom, it is proved that there exists a Sierpiński-Zygmund function, which is additive (i.e., is a solution of the Cauchy functional equation) and is absolutely nonmeasurable with respect to the class of all nonzero $\sigma$-finite diffused measures on the real line ${\mathbb R}$.