An outstanding problem is how to characterize the class of derivatives. Our purpose is to discuss the kinds of characterizations one might seek, to indicate the contributions of some attempts to characterize, to look at characterizations that are known, and to discuss the place of the class of derivatives among certain related classes of functions.

This paper investigates the family of continuous piecewise monotone functions which map a closed interval of the real line into itself. For these maps Preston \cite{1} and Blokh \cite{2} described the asymptotic behavior of the orbit of a “typical” point. Our results show that if the map is expanding on its intervals of monotonicity the dominant role is played by transitive \(f\)-cycles. Contrary to this for a “typical” map in a natural closure of the space of these maps there are no transitive \(f\)-cycles. Instead the behavior is dominated by the register shifts. This result is illustrated by an example.

In \cite[Thm 2.5]{CL:Unif} and \cite[Thm 2]{CL:Unif2} it was proved that there is no uniformly antisymmetric function with two- and three-element range by showing that \(K_3\) and \(K_4\) can be embedded into a graph \(G(h)\) (defined below) for all appropriate \(h\). In this note we will answer Problem 1 from \cite{CL:Unif2} by showing that under the continuum hypothesis there exists \(h\) for which \(K_5\) cannot be embedded into \(G(h)\). In particular, the technique used in the proof that there is no uniformly antisymmetric function with three-element range cannot be used for the four-element range proof. Whether there exists a uniformly antisymmetric function with a finite range remains an open problem. The notion of a uniformly anti-Schwartz function is also defined, and it is proved that there exists a uniformly anti-Schwartz function \(f\colon\mathbb{R}\to\mathbb{N}\).

The purpose of this paper is to study weakly continuous isotone bijections on a lattice of residuated maps (e.g. the space of probability distribution functions). We investigate the connection between weakly continuous isotone bijections and order automorphisms. Our final result gives necessary and sufficient conditions for a continuous isotone bijection to be an order automorphism.

It is known that a real-valued function on the unit interval belongs to Baire class one if and only if it is first return recoverable. Here it is shown that the trajectory used for recovering the function can be modified in such a way as to reflect local behavior of the function.

It is proved that the following conditions are equivalent: (a) \(f\) is an almost everywhere continuous function; (b) \(f=g+h\), where \(g,h\) are strongly quasi-continuous; and, (c) \(f=c+gh\), where \(c \in \mathbb{R}\) and \(g,h\) are s.q.c..

Recent research has shown that there is a significant cleavage between the structure of \(\omega\)-limit sets for continuous functions, and the structure of \(\omega\)-limit sets for Lipschitz functions. The development of these results rested on measure theoretic considerations. In this paper we show that there is no such divergence when one considers the topological structure of these two classes of \(\omega\)-limit sets. We show that an every nowhere dense compact set is homeomorphic to an \(\omega\)-limit set for a differentiable function with bounded derivative.

Let \((X,\rho)\) be a metric space, let \((\mathcal{K}(X),\widetilde{\rho})\) denote the space of non-empty compact subsets of \(X\) with the Hausdorff metric, and let \(E \subset X\). The purpose of this paper is to investigate the relationships between the entropy dimensions of \(E\) and of \(\mathcal{K}(E)\).

Some properties of \({\mathcal E}\)-continuous functions are investigated. In particular, the maximal family with respect to outer and inner compositions for the family of all \({\mathcal E}\)-continuous functions are described. Moreover, under some assumptions on \({\mathcal E}\) it is proved that every function \(f:\mathbb{R} \rightarrow \mathbb{R} \) can be represented as the composition of two \({\mathcal E}\)-continuous function. Similarly, every function \(f:\mathbb{R} \rightarrow \mathbb{R} \) can be represented as the limit of a transfinite sequence of \({\mathcal E}\)-continuous functions.

In a former paper ([Ju-No 1]) we introduced an axiomatic approach to the theory of non-absolutely convergent integrals in \(\mathbb{R}^n\). A specialization of this abstract concept leads to the well-behaved \(\nu (S)\)-integral over quite general sets \(A\) which yields the divergence theorem in its presently most general form. (See [Ju-No 3].) While the definition of the \(\nu (S)\)-integral is of descriptive type (i.e. in terms of an additive almost everywhere differentiable set function) we prove in this paper that it can equivalently be defined using Riemann sums. As an application we show that any function being variationally integrable over \(A\) in the sense of [Pf 3] is also \(\nu (S)\)-integrable over \(A\) and both integrals coincide.

Some properties describing the strong quasicontinuity of functions of one and two variables are considered.

Let \((Tf)(x)= x \hat f(x)\), where \(\hat f\) is the Fourier transform of \(f\). If \(P(t) = t \int^t_0 s^{-2} Q(s)\,ds\), \(t\gt 0\), where \(Q\) is some non-negative continuous function, then there is a constant \(A \gt 0\), such that \[ \int_{\mathbb{R}} Q(|Tf(x)|)\,\frac{dx}{x^2} \leq A \int_\mathbb{R} P(|f(x)|)\,dx\] holds. \(\!\)Moreover, \(\!\)if this inequality is satisfied for all \(f\), then \(t \!\int^t_0\! s^{-2} Q(s)ds \leq CP(t)\), \(t \gt 0\), for some constant \(C \gt 0\). Corresponding Orlicz space and dual inequalities for this and the Hardy averaging operator are also given.

In this article I prove that the difference of Darboux upper continuous functions possesses the Darboux property. Moreover I show that each Darboux function which is the difference of two (bounded) upper semicontinuous functions can be written as the difference of two (bounded) Darboux upper semicontinuous functions.

It is shown that under simple restrictions a series converges provided it’s set of terms has Kolmogorov dimension strictly smaller than \(\frac{1}{2}\).

It is proved that if a function is continuous on an interval, then its variations on a set taken relative to some covering relations are also continuous as interval-functions. This result is applied to a direct construction of continuous Perron major and minor functions of Henstock integrable function.

The paper is devoted to a topology of von Neumann type which is associated with a special invariant extension of the classical Lebesgue measure. The structure of the sets of density points for some sets having the Baire property with respect to this topology is investigated. In particular, a problem of Wilczynski concerning density points in the sense of category is solved.

For a real-valued function of several real variables that is \(n\)-times Peano Differentiable, a sufficient condition is given for the Peano derivatives of order \(n\) to be Baire*1. An immediate consequence will be that the order \(n\) Peano derivatives of an \((n+1)\)-times Peano differentiable function are Baire*1.

It is shown that an arbitrary fucntion \(f:\mathbb{R} \to \mathbb{R}\) can be written as the sum of two extendable connectivity functions.

Let \(\{\epsilon_n\}\) be a sequence of positive numbers converging to \(0\). For each \(n\) divide the unit interval \([0, 1]\) into sub-intervals of the type \([k \epsilon_n, (k+1) \epsilon_n)\), \(k = 0, 1, 2, \dots\) and define a function \(f^n\) as \(1, -1, 1, -1, \dots\) successively on these intervals. If \(\{f^{n_k}\}\) is any subsequence of \(\{f^n\}\) then the set of points at which \(\{f^{n_k}\}\) converges has Lebesgue measure zero. This is a generalization of the well known analogous result for Rademacher functions.

A relationship between the \(s\)-dimensional Hausdorff integral and the evolution with losses is established. Professor R. R. Nigmatullin showed that the evolution with loss can be described by a non-integer integral. This paper gives another way to describe the evolution. That is, the evolution can be expressed as Hausdorff integral.

Let \((X,d)\) be a separable metric space and \({\mathcal M}(X)\) the set of probability measures on the \(\sigma\)-algebra of Borel sets in \(X\). In this paper we will show that a function \(f\) is almost everywhere continuous with respect to \(\mu\in{\mathcal M}(X)\) if and only if \(\lim_{n\to\infty} \int_{X}f\,d\mu_n=\int_{X}f\,d\mu\), for all sequences \(\{\mu_n\}\) in \({\mathcal M}(X)\) such that \(\mu_n\) converges weakly to \(\mu\).

In this article two main results are proved. The first one is that each cliquish function \(f\colon \mathbb{R}^k \to \mathbb{R} \) is the sum of two quasi-continuous functions. It is also shown that we can moreover require that the summands preserve points of continuity of \(f\), are bounded provided that \(f\) is bounded and belong to the same class of Baire as \(f\) (if \(f\) is Borel measurable). The other main result is that each function \(f\colon \mathbb{R}^k \to \mathbb{R} \) which can be written as the product of finitely many quasi-continuous functions, can be expressed as the product of two quasi-continuous functions, and we can require that the factors belong to the same class of Baire as \(f\) (if \(f\) is Borel measurable).

We prove that each perfect linear set contains a perfect set which is (\(\Phi\))-uniformly symmetrically porous (Theorem 1). In the hyperspace of all nonempty compact sets in \([0,1]\) (endowed with the Hausdorff distance), the (\(\Phi\))-uniformly symmetrically porous nonempty compact sets form a \(G_\delta\) residual subspace (Theorem 2). We infer that the\linebreak (\(\Phi\))-uniformly symmetrically porous perfect sets form a \(G_\delta\) residual set in the space of all perfect sets in \([0,1]\) (Theorem 3).

Let \(f:\mathbb{R} ^+\rightarrow \mathbb{R} ^+\) and \((a_n)^\infty_{n=1}\) be a sequence of positive reals. We will say that \((a_n)^\infty_{n=1}\) is relatively \((R)\)-dense for \(f\) provided that for every \(x,y\in \mathbb{R} ^+\) with \(f(x)\lt f(y)\) there exists \(n,m\in\mathbb{N} \) such that \(f(x)\lt \frac{f(a_n)}{f(a_m)}\lt f(y)\). Sufficient conditions are given for a sequence of positive reals to be relatively \((R)\)-dense for certain functions.

We discuss some questions concerning the strengthened version of the Kuratowski-Ulam theorem obtained by Ceder. In particular, we refute Ceder’s conjecture that the measure analogue of his result holds. Further we consider mixed product \(\sigma\)-ideals \(\mathbb{K}\times \mathbb{L}\) and \(\mathbb{L}\times \mathbb{K}\) in \({\mathbb{R}}^2\) where \(\mathbb{K}\) and \(\mathbb{L}\) denote the families of meager and of Lebesgue null sets in \(\mathbb{R}\). For a set \(A\in \mathbb{K}\times \mathbb{L}\) (or \(A\in \mathbb{L} \times \mathbb{K}\)) we find large sets \(P\) and \(Q\) such that \(P\times Q\) misses \(A\). The proof is based on similar properties of \(\mathbb{K}\times\mathbb{K}\) and \(\mathbb{L}\times\mathbb{L}\) obtained by Ceder, Brodski\u i and Eggleston. A parametrized version of a Fubini-type theorem is also given.

It is well known that the fine topology in potential theory and the density topology in real analysis are not normal. This means that there exist pairs of disjoint finely (density) closed sets which cannot be separated by disjoint finely (density) open sets. A natural question arises about which pairs can be separated. We study those pairs of disjoint finely (density) closed sets which can be separated by disjoint finely (density) open sets. The key tool is the Lusin-Menchoff property of fine (density) topology. The main result is that finely (density) closed sets are finely (density) separated iff they are \(F_{\sigma }\)-“semiseparated” (Theorem 2.1, Theorem 2.2).

Every function \(f\colon [0,1]\to [-1,1]\) can be expressed as the sum of three extendable functions, as the maximum of two minimums of extendable functions and as the limit of a transfinite sequence of extendable functions.

In the spaces \(L^p_\mu[0,1]\), \(p\gt 1\) (which are Orlicz spaces of a very special kind), one has \(\| f\|_p=\sup\{\int^1_0fg\,d\mu :\int^1_0|g|^q\,d\mu \leq1\}=\inf\{t\gt 0 :\int^1_0|f/t|^p\,d\mu\leq1\}=(\int^1_0|f|^p\,d\mu)^{1/p}\), and any one of the three could serve as the definition for \(\| f\|_p\). For an arbitrary \(N\)-function, \(M\), the analogues of the first two of these formulations yield equivalent norms on the corresponding Orlicz space. On the other hand, the functional \(\rho_M\), defined on \(L^M_\mu[0,1]\) by \(\rho_M(f)=M^{-1}(\int^1_0M (f)\,d\mu)\), not only fails to be a norm in most cases, but frequently it is not even a reasonable approximation to the Orlicz norm.

We give an extension of Fatou’s lemma to tempered distributions \(\mathcal {S}'(G)\), where \(G\) is a homogeneous group.

In this paper, we extend the \({\mathcal{B}}\)-integral on \(\mathbb{R}\) to one on \(\mathbb{R}^{m}\). After proving a weaker version of the equiintegrability theorem, the equivalence of five generalized convergence theorems will be established, which include the equiintegrability theorem, two versions of the generalized controlled convergence theorem which are based on \(UACG^\nabla\) and \(UACG_{\mathcal{B}}^{\ast\ast}\), respectively, the generalized variational convergence theorem and the uniformly \(MGSRS_{\mathcal{B}}\) (modified \(GSRS\) with respect to \({\mathcal{B}}\)) convergence theorem.

A problem is posed concerning estimates on Green’s functions.