Studies comparing the classes of the continuous functions by different topologies were run in Łódź since the early seventies. The subject matter of these studies and the results achieved between 1973-1978 have been discussed in paper [8]. The present paper presents the results achieved in the years 1978-1995.

In this paper the class of improvable functions is defined and the basic properties of such functions is examined. Moreover, a necessary and sufficient condition under which a set \(A\) is the set of points of continuity of some \(\alpha\)-improvable discontinuous function is given and it is shown that the classes \(\mathcal{A}_\alpha\) and \(\mathcal{A}_\beta\) are different when \(\alpha \neq \beta\).

In this paper we shall introduce a notion analogous to the notion of homotopy and \(D\)-homotopy for connected transformations and present some theorems connected with this problem.

In this paper the classes \(\mathcal{A}_\alpha\) and the class \(\mathcal{A}\) are characterized and compared to the classes \(\mathcal{B}_1\), \(\mathcal{B}^*_1\) and the class of all Darboux functions.

A scattered set of real numbers can be exhibited in terms of endpoints of the components of some nondecreasing sequence of open sets (chain). We study this connection between scattered sets and chains providing characterizations of scattered sets, semiscattered sets, certain splattered sets and sets with countable closure. It is shown that any application of the Baire Category Theorem on the real line leads naturally to a chain of open sets and hence to an exceptional scattered set. Some applications of this fact are given.

Let \(\mathcal{F}\) be a family of real functions, \(\mathcal{F}\subseteq\mathbb{R}^\mathbb{R}\). In the paper we will examine the following question. For which families \(\mathcal{F}\subseteq\mathbb{R}^\mathbb{R}\) does there exist \(g\colon\mathbb{R}\to\mathbb{R}\) such that \(f+g\in\mathcal{F}\) for all \(f\in \mathcal{F}\)? More precisely, we will study a cardinal function \(\text{A}(\mathcal{F})\) defined as the smallest cardinality of a family \(F\subseteq\mathbb{R}^\mathbb{R}\) for which there is no such \(g\). We will prove that \(\text{A}(\text{Ext})=\text{A}(\text{PR})=\mathcal{c}^+\) and \(\text{A}(\text{PC})=2^{\mathcal{c}}\), where \(\text{Ext}\), \(\text{PR}\) and \(\text{PC}\) stand for the classes of extendable functions, functions with perfect road and peripherally continuous functions from \(\mathbb{R}\) into \(\mathbb{R}\), respectively. In particular, the equation \(\text{A}(\text{Ext})=\mathcal{c}^+\) immediately implies that every real function is a sum of two extendable functions. This solves a problem of Gibson \cite{Gib1}. We will also study the multiplicative analogue \(\text{M}(\mathcal{F})\) of the function \(\text{A}(\mathcal{F})\) and we prove that \(\text{M}(\text{Ext})=\text{M}(\text{PR})=2\) and \(\text{A}(\text{PC})=\mathcal{c}\). This article is a continuation of papers \cite{N2,CM,NR} in which functions \(\text{A}(\mathcal{F})\) and \(\text{M}(\mathcal{F})\) has been studied for the classes of almost continuous, connectivity and Darboux functions.

The aim of this paper is to demonstrate that several non-rigorous methods of mathematical reasoning in the field of divergent series, mostly related to the Euler and Hutton transforms, may be developed in a correct and consistent way by methods of nonstandard analysis.

In the paper we give and investigate the basic properties of a method for improving continuity, quasi-continuity and the Darboux property of real functions defined on a metric Baire space. This “improvement” is carried out with the use of Blumberg sets.

A theorem of Marcinkiewicz asserts that the Perron integrability of a function can be deduced from the existence of a single pair of continuous major and minor functions. We show that Perron-type integrals based on various symmetric derivatives do not have this property.

We define a generalized Riemann-Stieltjes integral with respect to charges (i.e., additive continuous functions on bounded \(BV\) sets), and study the linear space of all functions that are integrable with respect to every charge.

In this paper I examine products of a.e. continuous derivatives. Denote by \(\mathcal{N}_u\) the set of all \(x\) for which \(u (x) \neq 0\). First I prove that if \(u\) is a (bounded and/or non-negative) function with \(\mathcal{N}_u\) isolated, then \(u\) can be written as the product of two (bounded and/or non-negative) a.e. continuous derivatives. Next I show that if \(u\) is a.e. continuous and \(\mathcal{N}_u\) is a union of an isolated set and one with a null closure, then \(u\) can be written as the product of two a.e. continuous derivatives. I construct an example that we cannot require the factors be bounded in case \(u\) is bounded. Using this example I construct a bounded non-negative a.e. continuous function, \(v\), such that \(\mathcal{N}_v\) is the union of two isolated sets (so \(v\) is a Baire one star function and is the product of two bounded non-negative derivatives) and which cannot be written as the product of two a.e. continuous derivatives.

We obtain weighted Sobolev inequalities of mixed norm on parallelepipeds and then generalize them to product of domains satisfying the chain condition.

In the 1940’s, A. Denjoy proved that the typical point of a perfect nowhere dense set in \(\mathbb{R}\) is a point of strong porosity for that set. We prove two stronger versions of this for arbitrary metric spaces. Theorem 3 says that if \(E\) is any closed nowhere dense set in a metric space, and \(h\) is any porosity scale function, then the typical point in \(E\) is a point at which \(E\) is \(h\)-porous. Thus, if the metric space is complete, then “most” points of \(E\) are points at which \(E\) is very thin in the sense of porosity. Theorem 4 says that if \(F\) is closed and \(h\)-porous in \(\mathbb{R}\), then there exists a closed nowhere dense set \(E\) in \(\mathbb{R}\) containing \(F\) such that \(F\) is \(h\)-porous in the subspace \(E\). Therefore, in the sense of \(h\)-porosity, no nontrivial information about the porosity of a closed set in \(\mathbb{R}\) can be inferred from its porosity relative to some closed nowhere dense set in \(\mathbb{R}\).

We prove that if \(m\colon \mathcal{M} \to [0,+\infty]\) is an isometrically invariant \(\sigma\)-finite countably additive measure on \(\mathbb{R} ^{n}\), then there exists a countably additive isometrically invariant extension \(m'\colon\mathcal{M}' \to [0,+\infty]\) of \(m\) such that the canonical embedding \(e\colon{\mathcal{M}/ m} \to {\mathcal{M}'/ m'}\) of measure algebras defined by \(e([A]_{m})=[A]_{m'}\) is not surjective. This answers a question of Ciesielski \cite{C2}.

A conjecture of A. Kufner is verified, giving an approximate functional form of the Green’s function for a class of two-point boundary value problems. The result is applied to give a simple characterization of those weights for which the Hardy inequality for higher order derivatives is valid.

The paper deals with the space of all Denjoy-Perron integrable functions on a fixed interval endowed with the Alexiewicz norm and the completion of this space. The relatively weakly compact subsets of each space are characterized.

As an extension of a result from [F] an example is constructed of two different plane curves whose Crofton transforms are almost everywhere finite and equal each other on a metrically dense set.

We give a characterization of the set of nonwandering points of a continuous map \(f\) of the interval with zero topological entropy, attracted to a single (infinite) minimal set \(Q\). We show that such a map \(f\) can have a unique infinite minimal set \(Q\) and an infinite set \(B\subset\Omega (f)\setminus\omega (f)\) (of nonwandering points that are not \(\omega\)-limit points) attracted to \(Q\) and such that \(B\) has infinite intersections with infinitely many disjoint orbits of \(f\).

It is shown that the only Luxemburg functionals that satisfy a very simply formulated property are induced by \(p\)^th-power functions, \(0\lt p\lt \infty\). The known result that Orlicz spaces cannot be normed analogously to \(L_p\)-spaces follows as a consequence.

The notion of equi-derivatives is introduced and is compared with approximate equicontinuity. Moreover, it is proved that a function \(f\) of two variables whose sections \(f_x\) are equi-derivatives and sections \(f^y\) are measurable (derivatives) [have the Baire property] is measurable (a strong derivative) [has the Baire property].

In \cite{R} we introduced the notion of perfect measure zero sets and proved that every perfect measure zero set is permitted for any of the four families of trigonometric thin sets \(\\mathcal{N}\), \(\mathcal{A}\), \(\mathcal{N}_0\), and \(p\mathcal{D}\). Now we prove that the unions of less than \(\mathcal{t}\) perfect measure zero sets are permitted for the mentioned families. This strengthens a result of T. Bartoszyński and M. Scheepers \cite{BSch} saying that every set of cardinality less than \(\mathcal{t}\) is \(\mathcal{N}\)-permitted.

It is proved that the absolute continuity of the variational measure generated by an additive interval function \(F\) implies the differentiability almost everywhere of the function \(F\) and gives a full descriptive characterization of the Denjoy-Perron integral.

The aim of this paper is to describe in an explicit way all scrambled sets of chaotic weakly unimodal functions of type \(2^\infty\). For this purpose, we introduce in this setting a new coding method which turns out to be more useful here than the standard one from Milnor and Thurston's kneading theory \cite{MT}.

The dynamical system \((\mathcal{F}(X),T)\) which arises from an iterated function system \((X;w_1,\ldots ,w_m)\), where \(X\) is a compact metric space identified with the attractor of the system and the \(w_i\)’s are contractive invertible maps, is chaotic provided that the iterated function system satisfies the open set condition. The map \(T\) on the hyperspace \(\mathcal{F} (X)\) of the closed subsets of \(X\) is defined for a closed subset \(E\) as \begin{equation*} T(E)=w_1^{-1}(E)\cup \ldots \cup w_m^{-1}(E). \end{equation*} This extends results about the shift dynamical system for the non-overlapping case \cite{ba}.

A theorem of Grothendieck states that every closed subspace of the Banach space \(L^p(\mu)\), where \(\mu\) is a finite measure on a locally compact topological space, \(p \ge 1\), consisting of essentially bounded functions must have finite dimension. An analog of this result is proved concerning subspaces of the space of continuous functions on a compact metric space consisting of functions satisfying different Lipschitz-type conditions.

An elementary and natural method for demonstrating that certain exceptional sets are scattered is presented.

In this paper we show that the typical symmetrically continuous function and the typical symmetric function have \(c\)-dense sets of points of discontinuity. Also we show the existence of a nowhere symmetrically differentiable function and a nowhere quasi-smooth function by showing directly such functions are typical in the space of all real continuous functions.

We consider some additive properties of invariant \(\sigma\)-ideals and \(\sigma\)-algebras of subsets of the real line \({\mathbb{R}}\). In particular, we generalize two classical Sierpinski results, concerning additive properties of measure and category on \({\mathbb{R}}\), to a large class of invariant \(\sigma\)-ideals and \(\sigma\)-algebras.

A function \(f:\: \mathbb{R}\to\mathbb{R}\) preserves absolute convergence of series if for each absolutely convergent series \(\sum_{n=1}^{\infty} a_n\) its \(f\)-transform \(\sum_{n=1}^{\infty} f(a_n)\) is absolutely convergent. In this note, we shall study functions that preserve absolute convergence of series.

An \(s\)-set in Euclidean space is a set of finite, non-zero, Hausdorff \(s\)-dimensional measure. Call an \(s\)-set straight if its \(s\)-measure agrees with its Method I \(s\)-outer measure. Examples are given where there is a continuous, one-to-one function \(f\) on \(\mathbb{R}^n\) which is measure preserving on \( E\) so that \(f(E)\) is straight (such an \(f\) will be called a straightening of \(E\)). It is shown that any \(s\)-set can be written as a countable union of sets for which there are straightenings.

In this note we look at a subset of the metric space of metrics for an arbitrary set \(X\) and show that in terms of cardinality this can be very large while being extremely small in the porosity sense.

We consider some examples of invariant measures, defined on the real line, for which an analogue of the classical Steinhaus property does not hold.

In the paper it is shown that the strongly quasi-uniform limit of a sequence of 1-improvable discontinuous functions on a complete space \(X\) is a 1-improvable discontinuous function or a continuous function. Automatically the same result will be valid for uniform convergence.

In \cite{L14} (p. 224), P. Y. Lee and T. S. Chew use Corollary 1 of our paper essentially without proof, and without stating it explicitly, claiming that “it is easy to verify”. The same result is also used by P. Y. Lee in \cite{L13} (Theorem 10.2, p. 59). The aim of this article is to prove Corollary 1.

We investigate what large sets of real numbers such as uncountable, open, second Baire category, or positive Lebesgue measure contain copies of small sets such as finite, countable, or bounded. Some remarks on the related Erdös problem are included.

This paper gives a new proof of the equivalence of the McShane and Lebesgue integrals, which is independent of measure theory.

Recently Bartle [B], modifying a convergence theorem of Gordon, has established necessary and sufficient conditions for a Henstock integral convergence theorem. We point out that this result is not specific to the Henstock, or indeed to any definition of, an integral. Rather, it is a result about an arbitrary limit process and can thus be formulated in the setting of abstract convergence theory.

An iterated limits theorem is used to establish two sets of necessary and sufficient conditions for a Henstock integral convergence theorem. The motivation for the iterated limits theorem originates from Bartle’s notion of \(\gamma\)-convergence.

Some results about finite and countable unions of porous sets are established.

An example of a continuous function is given that is differentiable except on a countable set, but is not twice Peano differentiable on any nonempty, perfect set.

A monotone integral is given for scalar function, with respect to Riesz space values means, and also a necessary and sufficient condition to obtain a Radon-Nikodym density for two means.

We prove that, for every complete extension \(\mu\) of Lebesgue measure, the \(\mu\)-density topology is the Hashimoto topology generated by the density topology and the \(\sigma\)-ideal of \(\mu\)-null sets (cf. [1]).

An elementary but useful mean value theorem for integrals asserts that if \(f\) is a non-negative integrable function on \([a,b]\) and \(g\) is continuous there, then there is a \(\theta\in(a,b)\) such that \[g(\theta)\int^b_af(t)dt=\int^b_ag(t)f(t)dt.\] It does not seem to have been observed that this result has an equally useful extension to regulated \(g\), i.e, \(g\) which have right and left limits at each point.