Our paper was motivated by the desire to understand to what extend the notion of sequential functions is capable to construct self-affine functions and curves. We shall prove that each self-affine curve is generated by a sequential machine. This includes not only the curves obtained by the method of M. F. Dekking but also curves produced by iterated function systems. In order to expose the relations between the construction using sequential functions and the previous established methods we introduce several types of self-affine curves. Many classical examples will be discussed to illustrate the different types of self-affinity. In the general case, the realization of the curves by means of sequential functions is based on the classical Cantor representation of the real numbers in the unit interval. The combinatorial part of Dekking’s construction, the substitution, will appear as an ingredient of the sequential machine. The geometrical part can be recovered in the consistency condition for sequential machines, which assures that a sequential machine leads to a continuous function.

A function \(f:\mathbb{R}\to\mathbb{R}\) is said to have the intermediate value property provided that if \(p\) and \(q\) are real numbers such that \(p\neq q\) and \(f(p)\lt f(q)\), then for every \(y\in (f(p),f(q))\) there exists a number \(x\) between \(p\) and \(q\) with \(f(x)=y\). In 1875, G. Darboux showed that there exist functions with the intermediate value property that are not continuous \cite{22}. Because of his work with functions having the intermediate value property, these functions are called Darboux functions.

In this paper a Riemann scale of integrals is defined depending on the concept of sifting. Depending on the choice of sifting, the integral defined is equivalent to various classical integrals. The associated concepts of generalized absolute continuity are also defined and used to obtain descriptive definitions of the integrals introduced.

Let \(C\) and \(C'\) be Cantor sets in \(\mathbb{R}^n\) generated by Euclidean similarities, called clone Cantor sets. There are associated Hausdorff measures \(\mu_C\) and \(\mu_{C'}.\) We show that if there is a bi-Lipschitz map \(\phi\) of \(\mathbb{R}^n\) which maps \(C\) onto a clopen subset of \(C'\) then there exists a constant \(\lambda \gt 0\) and a subset \(A\) of \(C\) with \(\mu_C(A)\gt 0\) and such that for all \(\mu_C\)-measurable sets \(B\) of \(A\) we have \(\phi(B)\) is \(\mu_{C'}\)-measurable and \(\mu_{C'}(\phi B)=\lambda\mu_C(B)\). This result leads to an almost complete classification of clone Cantor sets up to bi-Lipschitz maps of Euclidean space.

In this paper we show that a Darboux function with finite variation, which is defined on closed, convex and boundary subset of \(\mathbb{R}^2\), can be extended to a Darboux function with finite variation, which is defined on \(\mathbb{R}^2\). Moreover, the set of all points of continuity and the set of all points of quasi-continuity for the first function are equal to the corresponding sets for the extension of this function.

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

Our Theorem 4.1 provides a new and direct proof of the measurability of Perron integrable functions, and hence of Kurzweil-Henstock integrable functions since both integrals coincide (cf. Theorem 11.5 and Theorem 11.6 in \cite{Go}, or the original paper of Kurzweil \cite{K}\h).

In this article we generalize Taylor’s theorem, using the local systems introduced by B. S. Thomson in \cite{T5}.

It is shown that the “standard double bubble” is the unique least-perimeter way to enclose and separate two given areas in the surface of the round sphere.

Sufficient conditions are given for a multifunction (set-valued function) to admit a continuously differentiable selection in one dimension. These conditions are given in terms of Clarke generalized gradients of the Hamiltonian associated with the multifunction.

We prove the Radon-Nikodým theorem for the Henstock integral and hence give a complete characterization of the primitive of a Henstock integrable function in Euclidean space.

The Banach-Zarecki Theorem states that \(VB \cap (N) = AC\) for continuous functions on a closed set. Hence it is a linear space. In this article we show that \(VB \cap (N)\) is a linear space on any real Borel set. Hence \(VBG \cap (N)\) will also be a real linear space for Borel measurable functions defined on an interval. As a consequence of this result, we show that the \(AK_N\) integral of Gordon (\cite{G14}) is well defined. We also give answers to Gordon’s questions in \cite{G14}.

In this paper we consider a cardinal invariant related to adding real functions defined on the real line. Let \(\mathcal{F}\) be such a family, we consider the smallest cardinality of a family \(\mathcal{G}\) of functions such that \(h+\mathcal{G}\) has non-empty intersection with \(\mathcal{F}\) for every function \(h\). We note that this cardinal is the additivity, a cardinal previously studied, of the compliment of \(\mathcal{F}\). Thus, we calculate the additivities of the compliments of various families of functions including the Darboux, almost continuous, extendable and perfect road functions. We briefly consider the general relationship between the additivity of a family and its compliment.

The author gives a general descriptive definition for integration, denoted by \({\mathcal P}\), which has as special cases the Lebesgue integral for bounded measurable functions, the Lebesgue integral, the Denjoy-Perron integral \({\mathcal D}^*\), the wide Denjoy integral \({\mathcal D}\), the Foran integral, the Iseki integral and the \(S{\mathcal F}\)-integral (\cite{Ene1}). This \({\mathcal P}\)-integral will admit Riesz type representation theorems (introducing an Alexiewicz norm, and identifying \(f\) with \(g\) whenever \(f = g\) \(a.e.\) on \([a,b]\)). The classical Riesz representation theorem for the linear and continuous functionals on \((C([a,b]),\|\cdot\|_\infty)\) is a consequence of Theorem 2. In addition it is shown that the space of \({\mathcal P}\)-integrable functions is of the first category in itself (see Section 5). Also a characterization of weak convergence on this space is given.

The Riemann integral can be defined in terms of upper and lower Riemann sums. In this paper, we show how the Henstock integral can also be defined in a similar manner.

In this paper refinements and extensions of properties that give generalized derivatives the basic properties of ordinary derivatives are discussed; for instance the Darboux, Baire-1, Denjoy, Zahorski properties.

The continuity of some functions of two variables equicontinuous in one variable is considered.

Let \(\mathcal{A}\) and \(\mathcal{B}\) be fields of subsets of a set \({\Omega}\), let \({\bf X}\) be a normed space with the Hahn-Banach extension property and let \({\mu: {\mathcal A}\rightarrow {\bf X}}\) and \({\nu: {\mathcal B} \rightarrow {\bf X}}\) be consistent, bounded, vector measures. We give necessary and sufficient conditions for \({\mu}\) and \({\nu}\) to have a bounded common extension to \({{\mathcal A} \vee {\mathcal B}}\), generalizing already known results for real valued charges.

Assume that \(G\subset \mathbb{R}^n\) is open and \(f:G\to\mathbb{R}\) is a differentiable function. C. E. Weil raised the gradient problem. In this problem it is asked whether \(\triangledown f\) satisfies the natural multidimensional generalization of the Denjoy-Clarkson property. We verify that if there are two dimensional counterexample functions to the gradient problem then their range should satisfy certain paradoxical convexity properties and the inverse image of ``many" values of \(\triangledown f\) is of positive linear measure.

Let \(\mathcal M\) stand for the space of all \(S\)-measurable real functions on the infinite \(\sigma\)-finite measure space \((X,S,\mu)\) endowed with the (metrizable but non-normable) topology of convergence in measure on sets of finite measure. Some natural subsets (including the \(L_p\)-spaces) are proved to be sigma-superporous in \(\mathcal M\). The possibility of finding non-sigma-porous meager sets in this non-normable setting is discussed.

The smoothing \(T_af\), for \(a \gt 0\), of a locally-integrable function \(f : \mathbb{R} \rightarrow \mathbb{R}\) is defined by \[(T_af)(x) := \frac{1}{2a} \int^{+a}_{-a} f(x+y) dy, \quad x \in \mathbb{R}.\] For a given \(g : \mathbb{R} \rightarrow \mathbb{R}\), any solution \(f\) of the equation \(T_af = g\) is called an unsmoothing of \(g\). In this note we analyse the problem of constructing a function \(\tilde{f} : \mathbb{R} \rightarrow \mathbb{R}\) such that \((T_a \tilde{f})(x_i) = g(x_i)\) for a given set of points \(x_1,x_2, \ldots , x_n \in \mathbb{R}\). We give an iterative process of constructing such an \(\tilde{f}\) under the assumption \(f \in L_2(\mathbb{R})\).

There is a general condition which guarantees that the limit in measure of the product is the product of the limits in measure.

We prove that for every real function \(f\) defined on a separable, complete and dense in itself metric space \(X\) there exists a \(c\)-dense set \(W\subset X\) such that \(f\upharpoonleft W\) is super quasi-continuous.

We show that there exists in ZFC an invariant extension of Lebesgue measure on \(\mathbb{R}\) such that for every uncountable subgroup \(H\) of \(\mathbb{R}\) there exists at least one selector of \(H\) measurable with respect to this measure. This answers a question of Sławomir Solecki in \cite{S}.

For a given space \(E\) equipped with a transformation group \(G\), the notions of a \(G\)-thick set and of a \(G\)-thin set are introduced and discussed. Some relationships between these notions and the theory of \(G\)-invariant (more generally, \(G\)-quasiinvariant) measures are considered.

The set of continuous functions of a single nonnegative real variable with at least one point of increase is a “small” set under two natural definitions: on the one hand, it is a set of first Baire category, and on the other hand, according to a famous result of Dvoretsky, Erdös and Kakutani, it is also a set with Wiener measure zero. In this paper, an analogous question for continuous functions of two nonnegative real variables is examined. Consider the set of continuous functions \(f\) on the nonnegative quadrant for which there exists a monotone curve along which the restriction of \(f\) has a point of increase. In this paper, it is shown that this set is “small” in the sense that it is of first Baire category. However, this set is “large” in the sense that it has full measure under the probability measure induced by the standard Brownian sheet.

The aim of this paper is to improve some recent results of Noiri and Popa on multifunctions by utilizing the concepts of almost \(\beta\)-continuous and weakly \(\alpha\)-continuous multifunctions. We give some more results on strong irresolvability.

It is well-known that for a given continuous function \(f\), \(f(0)=f(1)\) and for any natural number \(n\) there exist \(x_n\), \(y_n=x_n + 1/n\) such that \(f(x_n)=f(y_n)\). It is also known that if the graph of \(f\) (or more generally a planar curve connecting the point 0 and 1) does not have a horizontal chord of length \(a\) and \(b\) respectively then there is no horizontal chord of length \(a+b\) either (see \cite{YY}). It is almost immediate that the lengths of possible horizontal chords of \(f\) form a closed set \(F\) of the unit interval [0,1], and according to the remark above its complement \(G=[0,1]\setminus F\) is an additive set: \(a\in G\), \(b\in G\), \(a+b\le 1\) imply \(a+b \in G\). C. Ryll-Nardzewski, Z. Romanowicz and M. Morayne raised the problem whether this additive property is not just necessary but also sufficient for a set to be the complement of the chord-set of some continuous function. In this paper we answer their question affirmatively.

We show that there exists an absolutely convergent integral which properly includes the Lebesgue integral and that any such integral does not have positive property as a functional.