In this article we investigate the smallest (in the sense of inclusion) $\sigma$-field of subsets of R in which all functions of some families of functions from R to R satisfying some approximate quasicontinuity conditions introduced by Z. Grande in 1998 are measurable.

We characterize ordinary differentiability in terms on the harmonic derivative and a local Lipschitz type condition and apply the result to $C^{k,1}$-functions.

In this paper we prove that if $f:\R\times\R\to\R$ has a closed graph and all of its $x$-sections are continuous, and at least one $y$-section is continuous, then $f$ is continuous. It was already proved by Piotrowski and Wingler [PW] that if $f:\R\times\R\to\R$ has a closed graph and is separately continuous, then $f$ is continuous. Our result is stronger.

The best approximation operator by constants is extended from an Orlicz space $L^{\varphi}({\mathbb R}^m)$ to the space $L^{\varphi '} ({\mathbb R}^m),$ and some properties of this extended operator are established. Let $f_{\varepsilon}(x)$ be any best approximation of $f\in L^{\varphi '} ({\mathbb R}^m)$ on a suitable set $\beta_{\varepsilon} (X) \subset \mathbb{ R}^m.$ Weak and strong inequalities are proved for the maximal function associated with the family $\{f_{\varepsilon}(x)\}$ which are used in the study of pointwise convergence of $f_{\varepsilon}(x)$ to $f(x).$

Given a multifunction $F$ between topological spaces $X$ and $Y$, the reduced cluster function $C^r(F;\cdot): X \rightarrow 2^Y$ of $F$ is defined by $C^r(F;x)=\bigcap$ cl$(F(U\setminus \{x\}))$ running through the neighborhood system of $x$. By transfinite recursion, one defines iterated reduced cluster functions $C^{r,\alpha}(F;\cdot)$ for all ordinals $\alpha > 0$.

We characterize multifunctions $F$ that are invariant in the sense of $C^r(F;\cdot)=F$. For every countable ordinal $\alpha$, we describe the family of all iterated reduced cluster functions $C^{r,\alpha}(F;\cdot)$ of arbitrary multifunctions $F: X \rightarrow 2^Y$ and the family of all iterated reduced cluster functions $C^{r,\alpha}(f;\cdot)$ of arbitrary functions $f: X \rightarrow Y$, provided that $X$ and $Y$ are metrizable spaces and $Y$ is separable.

Let $\Omega\subset\mathbb{r}^n$ be a domain. A result of J. Kauhanen, P. Koskela and J. Malý in 1999 states that a function $f:\Omega\to\mathbb{R}$ with a derivative in the Lorentz space $L^{n,1}(\Omega,\mathbb{R}^n)$ is $n$-absolutely continuous in the sense of J. Malý. We give an example of an absolutely continuous function of two variables, whose derivative is not in $L^{2,1}$. The boundary behavior of $n$-absolutely continuous functions is also studied.

In 1958 T. M. Flett proved a theorem which is a variant of the Lagrange mean value theorem; namely, let $f:[ a,b] \to\mathbb{R}$ be a differentiable function in $[a,b]$ and $f^{\prime}( a)=f^{\prime}(b)$. Then there exists a number $\eta\in(a,b)$ such that $f(\eta)-f(a)=(\eta-a)\cdot f^{\prime}(\eta)$. Manav Das, Thomas Riedel and Prasanna K. Sahoo have given generalizations of Flett's theorem for approximately differentiable functions. Here we provide generalizations of these theorems for some local $\mathcal{S}$-systems.

In this article we introduce variable exponent Lebesgue spaces on metric measure spaces and consider a central tool in geometric analysis: the Hardy-Littlewood maximal operator. We show that the maximal operator is bounded provided the variable exponent satisfies a $\log$-H\"older type estimate. This condition is known to be essentially sharp in real Euclidean space, however, we show that this is not so in metric spaces.

Let $I$ be a nondegenerate interval and let $X\neq\emptyset$ be a set. For a function $f:X\times I \to {\mathbb{R}}$ and $x\in X$ define $v(x)$ as the total variation of the section $f_x$ on $I$. We investigate the regular dependence (measurability, Baire property, etc.) of $v$ on the regularity of the sections $f^t$.

We use the convergence notion of a run to define integration processes which subsume those of Riemann-Stieltjes, Lebesgue and Henstock-Kurzweil.

e study $\mathcal{I}$-convergence and $\mathcal{I}$-Cauchy sequences in metric spaces where $\mathcal{I}\subset\mathcal{P}(\mathbb{N}^k)$ is an ideal containing all singletons and $k\in\{1,2\}.$$

Let $SZ$, $D$, $Ext$, and $\overline{Ext}$ denote respectively the spaces of Sierpiński-Zygmund functions, Darboux functions, extendable connectivity functions, and uniform limits of sequences of extendable connectivity functions, with the metric of uniform convergence on them. We show that the subspaces $SZ \cap D$ and $SZ \cap \overline{Ext}$ are each porous in the space $SZ$, but $SZ \cap \overline{Ext}$ is not porous in the space $\overline{Ext}$. We also show that every real function can be expressed as a sum of two Sierpiński-Zygmund functions one of which belongs to $\overline{Ext}$. Ciesielski and Natkaniec showed in 1997 that if $\R$ is not the union of less than $\mathfrak c$-many nowhere dense subsets, then there exist Sierpiński-Zygmund bijections $f,g:\mathbb{R}\to\mathbb{R}$ such that $f^{-1} \notin SZ$ and $g^{-1}\in SZ$, but here we can additionally have $f$ and $g$ belonging to $\overline{Ext}$

In this paper, we show that nowhere monotone functions are the key ingredients to construction of continuous functions, absolutely continuous functions, and Lipschitz functions with large subdifferentials on the real line. Let $\partial_{c}f, \partial_{a}f$ denote the Clarke subdifferential and approximate subdifferential respectively. We construct absolutely continuous functions on $\R$ such that $\partial_{a}f=\partial_{c}f\equiv \R$. In the Banach space of continuous functions defined on $[0,1]$, denoted by $C[0,1]$, with the uniform norm, we show that there exists a residual and prevalent set $D\subset C[0,1]$ such that $\partial_{a}f=\partial_{c}f\equiv \R$ on $[0,1]$ for every $f\in D$. In the space of automorphisms we prove that most functions $f$ satisfy $\partial_{a}f=\partial_{c}f\equiv [0,+\infty)$ on $[0,1]$. The subdifferentiability of the Weierstrass function and the Cantor function are completely analyzed. Similar results for Lipschitz functions are also given.

Let $C([0,1])$ be the set of all continuous functions mapping the unit interval $[0,1]$ into itself, equipped with the metric $\rho$ of uniform convergence (and the induced topology $\tau$). A function $f\in C([0,1])$ is called Besicovitch if it is nowhere one-sided differentiable (finite or infinite). For the Lebesgue measure $\lambda$ we define the set $B(\lambda)\subset C([0,1])$ by $$B(\lambda)=\{f\vert~\forall~{\rm Borel}~A\subset [0,1]\colon~\lambda(A)=\lambda(f^{-1}(A))\text{ and }f \text{ is Besicovitch}\}.$$ We construct a set $X\subset B(\lambda)$ such that the space $(X,\tau\vert X)$ is homeomorphic to the product topological space $(\prod_{i=0}^{\infty}[0,1),\mu)$.

Four approximately continuous generalizations of the wide Denjoy integral are considered and a complete chart of relations between them is given.

n 1936, L. C. Young proved that the Riemann-Stieltjes integral$\int^b_a$ $f$ $dg$ exists, if $f\in BV_p,\,g\in BV_q, \frac{1}{p}+\frac{1}{q}>1$ and $f,g$ do not have common discontinuous points. In this note, using Henstock's approach, we prove that $\int^b_a$ $f$ $dg$ still exists without assuming the condition on discontinuous points. Some convergence theorems are also proved.

Let $f$ and $g$ be continuous real functions on the interval $[a,b]$, and let $K$ denote the set of all knot points of $f$. Let $E$ be a set of measure zero for which $f(E)$ has measure zero and $(f+g)(E)$ does not, and let $g$ be differentiable at each point of $E$ closure. We prove that $K$ must meet $E$, and moreover the intersection of $K$ with the closure of $E$ must contain a nonvoid perfect subset. Thus in particular, the function of Mazurkiewicz is a continuous N-Function with as many knot points as there are real numbers.

The paper is devoted to establishing a necessary condition for the existence of the Riemann-Stieltjes integral of continuous functions and simultaneously to finding a limit form of its relationship with the classical Riemann integral depending on a parameter. The mathematical technique for this study is based on changing the weighted first difference of a given function into the simple difference of a function determined by a first-order difference equation.

A function $f: \mathbb{R}^m \to \mathbb{R}$ is called $s_1$-strongly quasi-continuous at a point $\mathbf{x}\in \mathbb{R}^m$ if for each real $\varepsilon >0$ and for each set $A\ni {\bf x}$ belonging to the density topology, there is a nonempty open set $V$ such that $\emptyset \neq A \cap V \subset f^{-1}((f(\mathbf {x})- \varepsilon ,f(\mathbf {x})+ \varepsilon ))\cap C(f),$ denotes the set of continuity points of $f$. It is proved that every $\lambda $-almost everywhere continuous function $f:\mathbb{R}^m \to \mathbb{R}$ is the quasi-uniform limit of a sequence of $s_1 $-strongly quasi-continuous functions and that each measurable function $f:\mathbb{R}^m \to \mathbb{R}$ is the quasi-uniform limit of a sequence of approximately quasi-continuous functions $f:\mathbb{R}^m \to \mathbb{R}.$

In this paper we discuss integration by parts for several generalizations of the Riemann-Stieltjes integral. In addition, we obtain new results on integration by parts for the Henstock-Stieltjes integral and its interior modification for Banach space-valued functions.

Assuming that the union of fewer than continuumly many meager sets does not cover the real line, we construct an example of an additive almost continuous Sierpi{\'n}ski-Zygmund function which has a perfect road at each point but which does not have the Cantor intermediate value property.

We study several kinds of convergence of sequences of real numbers with indices $\overline{m} \in N^n$ and the respective versions for uniform and pointwise convergence of sequences of real functions. We obtain a theorem about generalized Baire classes of real functions. This theorem is related to a result by Katetov from 1972. We add some comments and applications connected with ideal convergence, a generalization of statistical convergence investigated by several authors.

The functional equation $k(p(x)) + k(x)= x, \ p(x)=x^2 +c$, was used to find quadratic invariant curves of a planar mapping. The continuity of its solutions $k$ on an interval is tied to its series representation through $\sum_{i=0}^{\infty}(p^{(2i)}(x)- p^{(2i+1)}(x))$, where the terms contain iterates of $p$. The intervals of convergence of the series deserve much attention. Because of the presence of iteration, such maximal intervals are sometimes difficult to determine. In this paper we show how numerical computations using Maple V5.1 and the use of discriminants and resultants may assist such development.

A characterization of when two sets in $\mathbb R$ can be separated by ambivalent sets is given. Two applications of the characterization are also presented.

The generalized Riemann (or Henstock) approach to integration is well-known for its explicitness and directness. It has been used to give an alternative definition to the Itô integral and the multiple stochastic integral, see [1,3,8,9,11,12,13,14]. In this paper we shall derive the Henstock-Fubini's Theorem for multiple stochastic integral based on the Henstock approach. We also show that the iterated multiple integral formula is a direct consequence of Henstock-Fubini's theorem.

It has been proved by Sierpiński that no linear basis of $\mathbb{R}$ over $\mathbb{Q}$ can be an analytic set. Here we show that the same assertion holds by replacing ``linear basis'' with ``transcendence basis''. Furthermore, it is demonstrated that purely transcendental subfields of $\mathbb{R}$ generated by Borel bases of the same cardinality are Borel isomorphic (as fields). Following Mauldin's arguments, we also indicate, for each ordinal $\alpha$ such that $1\leq \alpha\lt\omega_1$ ($2\leq \alpha\lt\omega_1$), the existence of subfields of $\mathbb{R}$ of exactly additive (multiplicative, ambiguous) class $\alpha$ in $\mathbb{R}$.

We give a new and simple proof that every polyharmonic function on $\R^n$ which is bounded is constant.

The dual space (with respect to the Alexiewicz norm) of the class of ${\cal BV}$-integrable functions on a compact cell ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i] \subset {\mathbb R}^m$ is shown to be isometrically isomorphic to the space of finite signed Borel measures on ${\mathop{\prod}\limits_{i=1}^{m}} [a_i, b_i)$, and the usual integral representation theorem holds. An example is also given to show that Lipschitz functions are not part of this dual space. This answers a question of Thierry De Pauw.

In a compact Hausdorff measure space we define an integral by partitions of the unity and prove that it is nonabsolutely convergent.

In the paper of Lee et al. an equivalent condition for a function $f$ to be of the first Baire class has been established. This condition is of an $\epsilon-\delta$ type, similarly as in Cauchy's definition of continuity of a function. In the first part of this paper we examine a problem whether it is possible to obtain other classes of functions by further modifications of the above condition. It turns out that, in some sense, the answer is negative. In the second part we consider a topological version of the condition of Lee et al.

The Radon transform is not uniquely defined for distributions. Moreover, on even dimensional Euclidean space, the formal integral defining the transform converges only for a subspace of tempered distributions.

In this paper we present four results of approximation of Carathéodory functions by the sequence of continuous functions. We obtain almost everywhere pointwise convergence with respect to the first variable, and discrete or uniform convergence on compact sets with respect to the second variable

It is proved that for a wide class of bases in ${\mathbb R}^m$ a function generates a $\sigma$-finite absolutely continuous variational measure if and only if this function belongs to $ACG_\delta$-class. It is also shown that under some additional assumptions on a basis, $\sigma$-finiteness follows from the absolute continuity of the variational measure.

In this paper, we study some properties of the Banach space $\beta_{\alpha}(X)$, which consists of all real Baire functions on a perfectly normal space $X$. We obtain the equality between Baire and Borel classes as a consequence of existence of an approximation property and a Tietze extension for these classes. Moreover, when $Y$ is a zero dimensional topological space, we obtain a refinement of the known results for the equality between ${\beta}_{\alpha}^{\circ}(X,Y)$ and $B_{\alpha}^{\circ}(X,Y)$.

Using results and methods of G. Choquet (1944) and M. Laczkovich and G. Petruska (1984), we slightly generalize their results on ``Whitney sets".

We note that in some models of ZFC the density topology can be not extraresolvable.

Let $m\geq 2$ be a natural number. Let $\nu _\lambda^{m} $ be the distribution of the random sum $\sum\limits_{n=0}^{\infty } \theta_n\lambda ^n$, where $\theta _n$ are i.i.d. and for every $n$ the random variable $\theta _n$ takes value in the set $\{0,\ldots,m-1\}$ with equal probabilities. As a generalization of Solomyak Theorem we prove that for Lebesgue a.e. $\lambda \in\left(1/m,1\right)$ the measure $\nu _\lambda^{m} $ is absolute continuous w.r.t. the Lebesgue measure. (For smaller $\lambda $, the measure $\nu _\lambda^{m}$ is supported by a Cantor-set, so if $\lambda\lt 1/m$ then $\nu _\lambda^{m}$ is singular.)