The paper deals with divergence phenomena for various approximation processes of analysis such as Fourier series, Lagrange interpolation, Walsh-Fourier series. We prove the existence of superdense (meaning residual, dense and uncountable) families of functions in appropriate function spaces over an interval $T\subset \mathbb R.$ One proves that for each function in the family, the corresponding approximation process is unboundedly divergent on a superdense subset of $T$ of full measure.

A real-valued function $f$ defined on an interval of $\mathbb{R}$ is said to be $n$-convex if all its $n$-th order divided differences are not negative. Let $f$ be such a function defined in a right neighborhood of $t_0 \in \mathbb{R}$ whose usual right derivatives, $f^{(r)}_+,$ $1\le r\le n,$ exist in that neighborhood and whose $(n+1)$-th order Peano derivative, $f_{n+1}(t_0)$, exists at $t_0$. Under these assumptions we prove that $f$ also possesses $(n+1)$-th order usual right derivative $f^{(n+1)}_+(t_0)$ at $t_0$. This result generalizes the known case for convex (that is 1-convex) functions. The latter appears in works of B. Jessen studying the curvature of convex curves and of J. M. Borwein, M. Fabian, D. Noll studying the second order differentiability of convex functions on abstract spaces.

Using the conditions increasing$^*$ and decreasing$^*$, and Thomson's variational measure, we give an easy proof of the Denjoy-Lusin-Saks Theorem [12, p.230]. In Theorem 5.1 we extend (the function is not assumed to be continuous) Thomson's Theorems 44.1 and 44.2 of [13], that are closely related to the Denjoy-Lusin-Saks Theorem. From this extension we obtain another classical result: the Denjoy-Young-Saks Theorem [5]. As consequences of the Denjoy-Lusin-Saks Theorem we obtain two well-known results due to de la Vallée Poussin [12, p. 125, 127]. Then wee extend these results (the set $E$ used there is not only Borel, but also Lebesgue measurable) and give in Theorem 8.1 a de la Vallée Poussin type theorem for $VB^*G$ functions, that is in fact an extension of a result of Thomson [13, Theorem 46.3]. Finally, we give characterizations for Lebesgue measurable functions that are $VB^*G \cap (N)$, and for measurable functions that are $VB^*G \cap N^{+\infty}$ on a Lebesgue measurable set.

We define symmetric continuity for functions defined on arbitrary subsets of $\mathbb{R}$. The main result is that when a symmetrically continuous function is defined on a measurable set (a set with the Baire property), then it is continuous almost everywhere (on a residual set, respectively). This generalizes the known result for functions defined on the whole real line.

Let $I$ and $J$ be \s-ideals on Polish spaces $X$ and $Y$, respectively. We say that the pair $\langle I,J\rangle$ has the Fubini Property (FP) if for every Borel subset $B$ of $X\times Y$, if all its sections $B_x= \{y\: \langle x,y\rangle\in B\}$ are in $J$, then its sections $B^y=\{x\: \langle x,y\rangle\in B\}$ are in $I$, for every $y$ outside a set from $J$. We study the question, which pairs of $\sigma$-ideals have the Fubini Property. We show, in particular, that:

-- $\langle$ MGR$(X), J\rangle$ satisfies FP, for every $J$ generated by any family of closed subsets of $Y$ (MGR$(X)$ is the $\sigma$-ideal of all meager subsets of $X$),

-- $\langle$ NULL$_\mu, J \rangle$ satisfies FP, whenever $J$ is generated by any of the following families of closed subsets of $Y$ (NULL$_mu$ is the $\sigma$-ideal of all subsets of $X$, having outer measure zero with respect to a Borel $\sigma$-finite continuous measure $\mu$ on $X$):

(i) all closed sets of cardinality $\leq 1$,

(ii) all compact sets,

(iii) all closed sets in NULL$_\nu$ for a Borel \s-finite continuous measure $\nu$ on $Y$,

(iv) all closed subsets of a $\hbox { $\mathbf{\Pi}^1_1$}$ set $A\subseteq Y$.

We also prove that $\langle$MGR$(X)$, MGR$(Y)\rangle$ and $\langle$ NULL$_\mu$, NULL$_\nu\rangle$ are essentially the only cases of FP in the class of \s-ideals obtained from MGR$(X)$ and NULL$_\mu$ by the operations of Borel isomorphism, product, extension and countable intersection.

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph. This answers a question of Gibson [15]. The second example is that of a connectivity function $F\colon\mathbb{R}^2\to\mathbb{R}$ with dense graph such that $F^{-1}(0)$ is contained in a countable union of straight lines. This easily implies the existence of an extendable function $f\colon\mathbb{R}\to\mathbb{R}$ with dense graph such that $f^{-1}(0)$ is countable. We also give a sufficient condition for a Darboux function $f\colon[0,1]\to[0,1]$ with a $G_\delta$ graph whose closure is bilaterally dense in itself to be quasi-continuous and extendable.

This paper presents one of many interesting aspects of relatively convergent series. Namely, given a sequence of elements of a Hilbert space we consider all possible ways of placing plus or minus signs in front of these elements to create an alternating sequence. In a convenient metric it makes sense to ask what is the `size' of the set of those choices of $+$ or $-$ for which the resulting series converges. The term `size' here refers to either Baire category or Lebesgue measure of this set. It turns out that especially the question of the Lebesgue measure of this set is quite intriguing and leads to interesting results generalizing known results for real-valued sequence.

This paper presents a study of typical series with alternating signs. Namely, given a sequence of real nonnegative numbers whose sum is infinity we consider all possible ways of placing plus or minus signs in front of each of these numbers. Choosing a convenient metric we ask what is the `size' (in terms of Baire category and porosity) of the set of those choices of $+$ or $-$ for which the resulting series converges. The author of this paper has studied this problem in his paper [D1] for the Euclidean metric. The main goal of this paper is to extend the results from [D1] for other standard metrics, such as the Fr\`echet or Baire metrics.

We shall use the ``iterated Venetian blind'' construction to show that the packing dimensions of plane sections of subsets of $\mathbb R^n$ can depend essentially on the directions of the planes. We shall also establish the instability of the packing dimension of sections under smooth diffeomorphisms.

In a harmonic space, the property that $k$ is a compact set of removable singularities for bounded harmonic functions defined in a neighborhood of $k$ is independent of the neighborhood chosen.

In this article we present a condition on the sections $f^y$ of a function $f: \mathbb{R}^2 \to \mathbb{R}$ having Lebesgue measurable sections $f_x$ and quasicontinuous sections $f^y$ which implies the measurability of $f$. This condition is more general than the Baire$^{**}_1$ property introduced by R. Pawlak in [7]. Some examples of quasicontinuous functions satisfying this condition and discontinuous on the sets of positive measure are given.

We consider the question of whether there is an orthonormal basis for $L^2$ consisting of continuous functions.

The paper includes a necessary and sufficient condition under which two $\psi$-density topologies generated by two functions $\psi_1$ and $\psi_2$ are equal. The condition is formulated in terms of the behavior of two sequences of sets: $A_k^+ =\{ x\in \mathbb{R}_+: \psi_1(2x) <\frac{1}{k} \psi_2(2x)\}$ and $B_k^+ =\{ x\in \mathbb{R}_+: \psi_2(2x) < \frac{1}{k} \psi_1(2x)\}.$

This note provides a reverse Hardy inequality associated with $BMO$-norm.

We prove that every real-valued $\mathcal{B}^*_1$ function $f$ defined on a~separable metric space $X$ is the sum of three quasicontinuous functions with closed graphs, and there is a $\mathcal{B}^*_1$ function which is not the sum of two quasicontinuous functions with closed graphs. Consequently, if $X$ is a separable metric space which is a Baire space in the strong sense, then the next three properties are equivalent: (1)$f$ is a $\mathcal{B}^*_1$ function, (2) $f$ is the sum of (at least) three quasicontinuous functions with closed graphs, and (3) $f$ is a piecewise continuous function.

We show that every almost continuous function $f\colon\mathbb{I}\to\mathbb{R}$ is also polygonally almost continuous. This solves a problem posed by Agronski, Ceder and Pearson (see [ACP])

This paper is devoted to relationships among various classes of $\cal I$-a.e. continuous functions (i.e., of functions whose sets of discontinuity points belong to certain $\sigma$-ideals $\cal I$ consisting of boundary sets). For instance, if $\cal K$ is the $\sigma$-ideal of first category sets and $\cal I$ denotes the $\sigma$-ideal of all sets that are: of Lebesque measure zero, $\sigma$-porous, or countable, then the set of $\cal I$-a.e. continuous functions is uniformly porous in the space of all $\cal K$-a.e. continuous Darboux functions from ${\mathbb R}^2$ into ${\mathbb R}^2$ equipped with the metric of uniform convergence. As a tool in the proofs, symmetric Cantor sets in ${\mathbb R}^2$ are used.

In this article we investigate some properties of discrete limits of sequences of functions satisfying some special approximate quasicontinuity conditions.

In the linear space$\mathcal{DP}[a,b]$ of all Denjoy-Perron integrable functions on an interval $[a,b]$ one wishes to introduce the most natural topology. Herein are some considerations that suggest what topology might be the most natural.

The problems of characterizing the class of universally polygonally approximable functions and the closure of that class are discussed.

In this work we provide a characterization of $C^{k,1}$ functions of one real variable (that is, $k$ times differentiable with locally Lipschitz $k$-th derivative) by means of $(k+1)$-th divided differences and Riemann derivatives. In particular we prove that the class of $C^{k,1}$ functions is equivalent to the class of functions with bounded $(k+1)$-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

Let $F\subseteq\mathbb{R}^{\mathbb{R}}$. The additivity of $F$, briefly A$(F)$, is the minimum cardinality of a family $G\subseteq\mathbb{R}^{\mathbb{R}}$ with the property that $h+G\subseteq F$ for no $h\in\mathbb{R}^{\mathbb{R}}$. In this paper we consider the notion of super-additivity which we will denote by $A^*$. If $F\subseteq\mathbb{R}^{\mathbb{R}}$, then $A^*(F)$ is the minimum cardinality of a family of functions $G$ with the property that for any $H\subseteq\mathbb{R}^{\mathbb{R}}$ if $|H|<A(F)$, there is a $g\in G$ such that $g+H\subseteq F$. We calculate the super-additivities of the families of Darboux-like functions and their complements.

We find order topologies that are universal for certain topological properties. An order topology $T$ enjoys a given property if and only if there is an order preserving homeomorphism of $T$ into the universal space for this property. We give similar results for order preserving mappings in place of homeomorphisms.

\noindent In this note, we prove that the shortest curve joining two points on a sphere in a real Hilbert space of dimension greater than $2$, lies in the great circle through these points. Our arguments will apply to all continuous curves on the sphere, not just to those that have tangent and principal normal vectors and curvature at each point.

We prove that the well-known Lagrange formula, the Darboux property and a classical result concerning the connected graph of a differentiable function are specific for ${\mathbb R}$, and surprisingly, the rule of L'H\^opital is also true for the vector case.

We define a singular point of a Henstock integrable function to be one which is not contained in any open interval on which the function is Lebesgue integrable. Then we give examples to illustrate the possible measure of the set of such singular points.

It will be shown that the density formula of [3] is proved by new method. As an application, the Hausdorff dimensions of and the Hausdorff measures of some Cantor-type sets will be evaluated.

Upper and lower limits for the length of the graph of one to one, onto, Baire one functions from the unit interval to itself are shown to be infinity and one, respectively. Such functions having graphs of large dimension and non-measurable functions are also considered.

The a.c. limits (introduced by Császár and Laczkovich) and the monotone limits of sequences of functions having everywhere finite unilateral limits are investigated.

The classical McShane integral has been generalized by D.H. Fremlin [2] to the case of an arbitrary $\sigma$-finite outer regular quasi-Radon measure space. We present an alternate approach to the Fremlin integral for a non-atomic, finite quasi-Radon space.

A function $f$ is metric preserving if for every metric space $(M,\rho)$ we have that $f \circ \rho$ is still a metric on $M$. In this article we look at the behavior of such functions with respect to continuity and differentiability. We include several pathological examples and some open questions.

TThe note presents a new approach to the notion of continuity of real function at a point. It is applied to obtain a characterization of continuity at a point with respect to *-topology (Hashimoto topology), density topology and $I$-density topology (Wilczyński topology). The latter is closely related to the definition of density point of measurable set formulated by W. Wilczyński in [8].

In this note we prove that for a quite large class of topological spaces every upper semi-continuous function, which is a discrete limit of continuous functions, it is also a pointwise decreasing discrete limit of continuous functions. This question was motivated by a paper of Zbigniew Grande. He asked that whether it be true for the topology of right hand continuity on the real line. He gave a partial answer showing that under some additional condition imposed on the function the answer is affirmative.

Two theorems on symmetry properties of cluster sets relative to a grill for the domain of the functions are proved here. One of these contains a result of Young [8] and its analogue for qualitative and other cluster sets. (Wilczynski [7] introduced the notion of qualitative cluster sets.) The other contains a result of Erdös and Piranian [2], one of Dolzhenko [1], and analogous results for other cluster sets

In [3] we proved that generalized Carathéodory's conditions (the so called (G) conditions) introduced in [6], imply well-known general conditions (the so called (CL) conditions) which guarantee existence and some properties of solutions of the Cauchy problem, in the Carathéodory sense. In this paper we prove that these two conditions are equivalent.

When it is known that $\int_a^bf_n\to\int_a^b f$ for a sequence of Henstock integrable functions $\{f_n\}$ we give necessary and sufficient conditions for $\int_a^b f_n\,g_n\to\int_a^b f\,g$ for all convergent sequences $\{g_n\}$ of functions of uniform bounded variation. The conditions are easy to apply and involve either the uniform boundedness or uniform convergence of the indefinite integrals of $f_n$. The proof uses Stieltjes integrals and applies to bounded or unbounded intervals on the real line. It is shown how to define Stieltjes integrals on unbounded intervals without treating them as improper integrals. The special cases $f_n\equiv f$ or $g_n\equiv g$ are also examined. The Abel and Dirichlet tests for integrability of a product are obtained as corollaries as well as a form of the Riemann-Lebesgue lemma. And, if $\Phi : \mathbb{N}\to (0,\infty)$ it is shown what conditions on $\{f_n\}$ and $\{g_n\}$ give $\int_a^b f_n\,g_n=O(\Phi(n))$ as $n\to\infty$.

We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil \cite{Kur} and Henstock \cite{Hen}. More precisely, the presentation is close to the McShane approach \cite{McS}. Besides its simplicity of presentation, four advantages characterize this theory: %{\leftskip=1cm \item{(i)} the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces; \item{(ii)} when a function is integrable its norm is also integrable, and the proof is straightforward from the definition; \item{(iii)} for finite dimension spaces the theory is equivalent to the McShane's theory, which is itself equivalent to the Lebesgue's theory; \item{(iv)} and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. \par%}

For any group $G$, the notion of an invariant measurable structure on $G$ is introduced. The following question is investigated: does there exist a subgroup of $G$ nonmeasurable with respect to this structure? It is demonstrated that, for an uncountable solvable group $G$, such a subgroup of $G$ always exists.

The purpose of this paper is to study a set of convergence points of sequence of real functions from a given class. Here, continuous func\-tions, Borel measurable functions, approximately continuous functions and derivatives are considered.

We construct a continuous strictly increasing function such that at each point one of its right Dini derivates is $0$ or $\infty$, and at each point one of its left Dini derivates is $0$ or $\infty$. Thus at no point can it have a positive real unilateral derivative.

We show that typical continuous functions of the form $f:M \to M$, where $M$ is a compact metric space with the fixed-point property and the absolute-retract property, are not chaotic in the sense of Devaney. Typical continuous functions on the compact interval have been shown to be chaotic in terms of other definitions of chaos. Results are also presented concerning the chain recurrent set for typical continuous functions and concerning functions for which the chain recurrent set is the entire space.

In this query we ask if the following 11 classical theorems on convergence are equivalent: the Lebesgue-Beppo Levi Theorem \cite[p. 141]{N1}, a theorem on the integration of a series with positive terms \cite[p. 142]{N1}, the Fatou Lemma I \cite[p. 172]{H12}, the Fatou Lemma II \cite[p. 140]{N1}, the Fatou Lemma III \cite[p. 140]{N1}, Lebesgue's Dominated Convergence Theorem I \cite[p. 172]{H12}, Lebesgue's Dominated Convergence Theorem II \cite[p. 173]{H12}, Lebesgue's Dominated Convergence Theorem III \cite[pp. 149-50]{N1}, Vitali's Theorem \cite[p. 152]{N1}, Lebesgue's Dominated Convergence Theorem for Bounded Functions I \cite[p. 127]{N1}, Lebesgue's Dominated Convergence Theorem for Bounded Functions II.