487-496: During the 14th Summer School on Real Functions Theory in Liptovsky Ján, Slovakia in 1996, Richard G. (Jerry) Gibson gave a talk concerning Darboux-like functions. Following his talk we wrote the survey article Darboux-like functions, that was published in the Real Analysis Exchange. In that paper we collected not only known facts on the subject but also presented some open problems. In the last two years a considerable number of papers have been written that solve some of the open questions posed in our survey article. In this short abstract we would like to present those solutions and to list the problems that remain open. This paper is based on the talk given by Tomasz (Tomek) Natkaniec at the 15th Summer School on Real Functions Theory in Liptovsky Ján, Slovakia, in 1998. Most of the new results are contained as a part of the Polish-American project it Set Theoretic Analysis and has been described in K. Ciesielski's survey Set Theoretic Real Analysis. You can find these results on the Set Theoretic Analysis web page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html

It is consistent that CH fails and every function which is the pointwise limit of an $\omega_2$-sequence of Baire-2 functions is Baire-2. It is also consistent that CH fails and there is a function which is not such a limit.

We prove that if $f$ is a continuous interval map such that all wandering intervals converge to periodic orbits, then the family of periodic orbits is dense in $\omega$-limit sets with Hausdorff metric.

For a local system ${\mathcal S}$ and a function $f:{\mathbb R} \to {\mathbb R}$, Thomson defines in \cite{T5} the metric outer measure ${\mathcal S}\text{-}\mu_f$ on ${\mathbb R}$, and in \cite{Ene1}, we introduce the properties ${\mathcal S}AC$ (${\mathcal S}$-absolute continuity) and ${\mathcal S}ACG$. In this paper we show that for some particular local systems, such as ${\mathcal S}_o$, ${\mathcal S}_{ap}$ and ${\mathcal S}_{\alpha,\beta}$, there is a strong relationship between ${\mathcal S}\text{-}\mu_f$ and ${\mathcal S}ACG$. For example: \emph{a function $f:[a,b] \to {\mathbb R}$ is ${\mathcal S}_oACG$ on a Lebesgue measurable set $E \subset [a,b]$ $\Longleftrightarrow$ ${\mathcal S}_o\text{-}\mu_f$ is absolutely continuous on $E$ $\Longleftrightarrow$ $f$ is $VB^*G \cap (N)$ on $E$ and $f$ is continuous at each point of $E$} (see Theorem 8.4). In Theorem 5.1, we show that \emph{a function $f:[a,b] \to {\mathbb R}$, Lebesgue measurable on $E \subseteq [a,b]$, is ${\mathcal S}_{ap}ACG$ on $E$ if and only if ${\mathcal S}_{ap}\text{-}\mu_f$ is absolutely continuous on $E$}. This result remains true if ${\mathcal S}_{ap}$ is replaced by ${\mathcal S}_{\alpha,\beta}$ with $\alpha,\beta \in (1/2,1]$ (see Theorem 5.1). We also give several characterizations of the $VB^*G$ functions on an arbitrary set, that are continuous at each point of that set (see Theorem 7.4).

Results concerning extension of submultiplicativity and supermultiplicativity for Orlicz functions are proved. A typical result is the $\text{following}$: If the Orlicz function $\varphi$ is submultiplicative at infinity, then an Orlicz function $\psi$, which is submultiplicative on ${\mathbb R}_+$, equivalent to $\varphi$ at infinity and satisfying $\psi(u)/u \to 0$ as $u \to 0$ exists if and only if the conjugate function $\varphi^*$ satisfies the $\Delta_2$-condition at infinity. Some complementary results and (counter-)examples are also included.

We find variations of the classical theorem that any nonzero real ring homomorphism, $u$, of $C(X)$, for a compact Hausdorff space, $X$, is fixed. We let $X$ be a locally compact Hausdorff space and we let $u$ be defined on certain subrings of $C(X)>$ We also vary the hypothesis on $u$ in other ways.

We study the maximal additive, multiplicative and lattice-like families for the class of all extendable functions. This article is a continuation of earlier papers, in which the same questions concerning other Darboux-like functions have been studied.

We describe here an example of a Darboux function $k$ from the unit interval $\mathcal{I}=[0,1]$ onto itself such that $k$ is not the composition of any finite collection of connectivity functions from $\mathcal{I}$ into $\mathcal{I}$. This answers a question of Ceder [2].

The main result of this paper states the following. For each natural number i, let $G_i$ be a proper additive subgroup of the reals, $A_i$ a set that contains no arithmetic progression of length three, $H_i$ a basis for the vector space $\R$ over the field of rationals, and $E^{+}(H_i)$ the set of all finite linear combinations from the elements of $H_i$ with nonnegative rational coefficients. Then the complement of a finite union of sets $G_i\cup A_i\cup E^{+}(H_i)$ is everywhere of second category. We also prove that the complement of a union of fewer than continuum many translates of sets that have distinct distances is everywhere of second category.

The goal of this note is to construct a uniformly antisymmetric function $f\colon\mathbb{R}\to\mathbb{R}$ with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski's survey [5].) A problem of existence of uniformly antisymmetric function $f\colon\mathbb{R}\to\mathbb{R}$ with finite range remains open.

In this paper we consider the properties of ideals of some rings of almost continuous functions, being extensions of rings of continuous functions.

We prove that the set of all continuous mappings of $[0,1]^n$ to ${R}^n$ with Luzin's property (N) with respect to Lebesgue measure is a coanalytic non-Borel and first category subset of the space of all continuous mappings. Some generalizations, e.g. to cases of other Radon or Hausdorff measures are given.

We investigate the possibility of making all functions in Baire class $\alpha$, $\alpha<\omega_1$, close to continuous by adding to them a single function of Baire class $\alpha+1$.

Let $\mathcal{N}$ denote the ideal of null sets in $\mathbb{R} .$ Using CH Sierpiński constructed a set $A\subset \mathbb{R}$ satisfying $A\notin \mathcal{N} , \mathbb{R} \setminus A \notin \mathcal{N} ,$ and $(A+h)\setminus A \in \mathcal{N}$ for every $h\in \mathbb{R} .$ He also constructed a set $H\subset \mathbb{R}$ such that each horizontal section of $H$ and the complement of each vertical section of $H$ belong to $\mathcal{N} .$ In this note we investigate the existence of sets with analogous properties when $\mathcal{N}$ is replaced by an arbitrary invariant ideal. We also discuss the relationship among several related statements, including some inequalities between the cardinal invariants of the ideal in question. In the case of null sets in $\mathbb{R}$ we show that the nonexistence of $A\subset \mathbb{R}$ with $A\notin \mathcal{N}, \ \mathbb{R} \setminus A \notin \mathcal{N} ,$ and $(A+h)\setminus A \in \mathcal{N}\ (h\in \mathbb{R})$ is equivalent to the difference property of the class $\mathcal {L}$ of Lebesgue measurable functions defined on $\mathbb{R} .$ As an application we obtain that the difference property of the class $\mathcal{L}$ is consistent with ZFC.

A function $f$ on the real line is rearrangeable if there exists a bijection $\gamma$ on its domain such that $f\circ\gamma$ is continuous. Combinatorial and analytic aspects of the problem are investigated.

Two new notions of separate continuity either of which is equivalent to continuity are introduced.

The paper is an investigation of the collection of absolutely measurable functions defined on compact, connected manifolds. Several analytical properties of these functions defined on the manifold $I$, the unit interval of $\mathbb R$, have been studied by C. Goffman, D. Waterman and the author in Homeomorphisms in analysis [Math. Surveys Monogr., Number 54, American Mathematical Society, Providence, 1997]. It will be shown that these properties also hold for all compact, connected manifolds. The method of proof differs from those used earlier for the interval $I$. The key element here is the use of the von Neumann-Ulam-Oxtoby Theorem for compact connected manifolds (proved here for the first time) which concerns measures induced by homeomorphisms.

The additivity $A(\mathcal{F})$ of a family $\mathcal{F}\subseteq\mathbb{R}^{\mathbb{R}}$ is the minimum cardinality of a $G\subseteq\mathbb{R}^\mathbb{R}$ with the property that $f+G\subseteq\mathcal{F}$ for no $f\in\mathbb{R}^{\mathbb{R}}$. The values of $A$ have been calculated for many families of Darboux-like functions in $\mathbb{R}^\mathbb{R}$. We extend these results to include some families of Darboux-like functions in $\mathbb{R}^{\mathbb{R}^n}$. To do this we must define $(n,k)$-additivity which is much more flexible than additivity.

Necessary and sufficient conditions on a measure are obtained for the corresponding maximal operators to be of reverse weak $(1,1)$ type.

Let $\mathcal{K}$ be the class of compact subsets of $I=[0,1]$, and $\mathcal{K}^{*}$ consist of the nonempty closed subsets of $\mathcal{K}$. We study the maps $\Lambda :C(I,I)\rightarrow \mathcal{K}$ and $\Omega :C(I,I)\rightarrow \mathcal{K}^{*}$ defined so that $\Lambda (f)$ is the set of $\omega $-limit points of $f$, and $\Omega \bigl( f\bigr) $ is the collection of $\omega $-limit sets of $f$. We find that, in general, neither map is continuous. We do get more positive results, however, if we restrict ourselves to certain types of $\omega $-limit sets and better behaved classes of functions. We find that when $f$ has only a finite number of $\omega $-limit sets, each demonstrating a certain type of stability, the function $\Omega $ is continuous at $f$. The map $\Omega \mid \mathcal{E}$ is also studied, where $\mathcal{E}$ is composed of those continuous functions with zero topological entropy, and a significant degree of stability of $\Omega \mid \mathcal{E}$ is established.

Alikhani-Koopaei has recently conjectured that two commuting continuous functions typically share no periodic points. After discussing the history behind Alikhani-Koopaei's conjecture, we use the Baire Category Theorem to investigate the likelihood that two commuting continuous functions have disjoint sets of periodic points, as well as the structure of the set $\mathcal{F}=\{g\in C(I,I):gf=fg\}$. We then turn our attention to the iterative properties possessed by commuting pairs of continuous functions.

In this note some properties of the quasi-uniform convergence of [9] studied and related to the quasi-continuous functions of [5] in terms of the entourages [1]). Then, quasi-uniform convergence is investigated as a convergence structure in the sense of [6]. Also, quasi-uniform and quasi-normal convergences of sequences of real-valued functions from [2], [3] are compared.

It is shown that for Banach--space--valued functions the variational Henstock integral is equivalent to the Henstock integral if and only if the range space is of a finite dimension. The same is true for the equivalence of the variational McShane integral and the McShane integral.

We show that every set of reals is a set of points of weak symmetrical continuity for some function and compare it with other generalized continuities. We also make some remarks on points of weak symmetric continuity when function is finite valued.

A generalized conditional expectation and the corresponding martingale is defined in terms of the Kolmogorov $A$-integral. It is proved that the uniform $A$-integrability of a martingale sequence is a sufficient condition for the sequence to be closed on the right by the $A$-integrable last element.

Let $I=[0,1]$. We show that those functions in $C(I,I)$ possessing zero topological entropy form a nowhere dense perfect subset of the continuous self maps of the interval. We also show that every function with zero topological entropy that possesses an infinite $\omega $-limit set is the uniform limit of functions having only finite $\omega $-limit sets.

We construct (in ZFC) an example of Sierpiński--Zygmund function having the Cantor intermediate value property and observe that every such function does not have the strong Cantor intermediate value property, which solves the problem of R.Gibson [8, Question 2]. Moreover we prove that both families: $SCIVP$ functions and $CIVP\setminus SCIVP$ functions are $2^{\mathfrak{c}}$ dense in the uniform closure of the class of $CIVP$ functions. We show also that if the real line $\mathbb{R}$ is not a union of less than continuum many its meager subsets, then there exists an almost continuous Sierpiński--Zygmund function having the Cantor intermediate value property. Because such a function does not have the strong Cantor intermediate value property, it is not extendable. This solves another problem of Gibson[8, Question 3]{RG}.

An elementary proof is given of the existence of a measure-preserving bijection of $\mathbb R^n$ that maps a preassigned Borel set with Lebesgue measure~$1$ onto the unit cube. The proof requires the use of only the Vitali Covering Theorem, translations and elementary properties of infinite sets.

We construct a compact subset of $\R$ with Hausdorff dimension 1 that intersects each of its non-identical translates in at most one point. Moreover, one can make the set to be linearly independent over the rationals.

Recently several authors have established a remarkable property of the variational measures associated with a function. Expressed in classical language, this property asserts that if a function is ACG${}_*$ on all sets of Lebesgue measure zero then the function must be globally ACG${}_*$. This article is an exposition of some ideas related to this property with the intention of bringing it to the attention of a wider audience than these original papers might attract.

Let $X$ be a complete separable metric space, let $(s)$ be the set of all Marczewski \cite{sm} measurable subsets of $X$, and let $(s^0)$ be the the set of all Marczewski null subsets of $X$. It is already known that $(s)/(s^0)$ is a complete Boolean algebra, but the known proofs of this involve complicated preliminaries. We present a simple proof that $(s)/(s^0)$ is a complete Boolean algebra.

We prove that in an at least three dimensional Euclidean space the Dynkin system generated by the family of all open balls with radii at least one (that is, the smallest collection containing the open balls with radii at least one and closed under complements and countable disjoint unions) does not contain all Borel sets. We also give a simple characterization of the sets of this Dynkin system.

The well-known Hake Theorem asserts that if a function $f$ is Denjoy$^*$ integrable then it is also Perron integrable, and the two integrals are equal. In \cite{Ene1} we introduced a very strong Perron integration $({\mathcal P}_{1,1})$ and proved the corresponding Hake-type theorem, using the Vitali-Carath\'eodory Theorem. In this paper we give a new, less technical proof of this result, using essentially Lusin's Theorem.

A very short proof of an inequality due to Cutler and Olsen is presented.

We characterize the coordinate functions of Hilbert's space-filling curve using a directed-graph iterated function system and use this to analyze their fractal properties. In particular, we show that both coordinate functions have graphs of Hausdorff dimension $\frac{3}{2}$ and level sets of dimension $\frac{1}{2}$.

In \cite{F17}, Faure and Gu\'enard put the following problem: Characterize the Denjoy$^*$-integrable functions $f:[a,b] \to \overline{{\mathbb R}}$ that can be approximated by two Baire $1$ functions $g_\epsilon$ and $h_\epsilon$, $\epsilon > 0$, that are ${\mathcal D}^*$-integrable. In the present article we show that this class of functions coincides with the class of all ${\mathcal D}^*$-integrable functions $f:[a,b] \to \overline{{\mathbb R}}$.