Editors Note. This paper was originally given as the Gibson Lecture at the 32nd Summer Symposium in Real Analysis held at Chicago State University, June 10-14, 2008. It is being published here as a survey at the request of the Editor and with the permission of the author.

To the Gauss-like continued fraction expansions we associate a conformal iterated function system whose limit set is of Lebesgue measure equal to 1. We show that the Texan Conjecture holds; i.e. for every $t \in [0,1]$ there exists a subsystem whose limit set has Hausdorff dimension equal to $t$.

We give a new characterization of the Baire class $1$ functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions, which are particularly simple Lipschitz functions. Moreover we highlight the link between the two classical stratifications of the Borel functions by showing that the Baire class functions of some level are exactly those obtained as uniform limits of sequences of Delta functions of a corresponding level.

In this paper we introduce the concept of $\;(p,\alpha)$-bounded variation which generalizes the Riesz p-variation. The following result is proved: a function $\; f:[a,b] \rightarrow \mathbb{R}\;$ is of $\; (p,\alpha)$-bounded variation $\;(1<p<\infty)\;$ if and only if $f$ is $\alpha$-absolutely continuous on $[a,b]$ and $f^{^{\prime}}_{\alpha} \in L_{(p,\alpha)} [a,b]$. Moreover it is shown that the $\;(p,\alpha)$-bounded variation of a function $f$ on $[a,b]$ is given by \begin{equation*} V_{(p,\alpha)} (f) = \|f^{^{\prime}}_{\alpha}\|^{p}_{L_{(p,\alpha)} [a,b]}. \end{equation*}

Let $\tau$ be a topology on the real numbers which is finer than the usual topology such that $\R_\tau$ is a weak $P$-space. In this paper, conditional completeness and conditional $\sigma$-completeness of $C(X)$ and $C(X,\R_\tau)$ are compared. In particular, it is shown for zero-dimensional spaces that $C(X,\R_\tau)$ is conditionally $\sigma$-complete if and only if $X$ is a $P$-space and that $C(X,\R_\tau)$ is conditionally complete if and only if $X$ is an extremally disconnected $P$-space.

A measure $\mu$ on $\mathbb{R}^n$ is called locally and uniformly $h$-dimensional if $\mu(B_r(x))\leq h(r)$ for all $x \in \mathbb{R}^n$ and for all $0 <r < 1 $, where $h$ is a real valued function. If $f\in L^2(\mu)$ and $\mathcal{F}_\mu f$ denotes its Fourier transform with respect to $\mu$, it is not true (in general) that s. However in this paper we prove that, under certain hypothesis on $h$, for any $f\in L^2(\mu)$ the $L^2$-norm of its Fourier transform restricted to a ball of radius $r$ has the same order of growth as $r^n h(r^{-1})$ when $r\to\infty$. Moreover we prove that the ratio between these quantities is bounded by the $L^2(\mu)$-norm of $f$ (Theorem 3.2). By imposing certain restrictions on the measure $\mu$, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which $h(x)=x^\alpha$.

In [9] G. F. Stefansson has studied the Banach space $L_1(\nu, X, Y)$, the space of all tensor integrable functions $f : \Omega \to X $ with respect to a countably additive vector valued measure $\nu : \to \Sigma \to Y$ and also the tensor integral of weakly $\nu$-measurable functions. In [1] we obtained some Banach space properties of $L_1(\nu, X, Y)$ and also of w-$L_1(\nu, X, Y)$, the space of all weakly tensor integrable functions. In the present paper, for $1 < p < \infty$, we define the spaces $L_p(\nu, X, Y)$ and w-$L_p(\nu, X, Y)$ of all $\check \otimes_p$-integrable functions and weakly $\check \otimes_p$-integrable functions respectively and discuss several basic properties of these spaces. We also study vector measure duality in $L_p(\nu, X, Y)$ for $1 < p < \infty$.

We consider sets of generalized discontinuity of real functions with respect to local systems fulfilling the intersection condition. We give a sufficient condition for countability of such set. This result is used to prove its $\mathbb{I}$-density analogue.

In this paper we present a discussion of the relations of the classes of Darboux-like functions within the classes of Hamel functions and Sierpńiski-Zygmund Hamel functions. We prove that the inclusion relations among Darboux-like classes remain valid in both cases (under the assumption of CH for Sierpiński-Zygmund Hamel functions). In particular, assuming CH we prove the existence of a Sierpiński-Zygmund Hamel function which is connectivity but not almost continuous. In addition, we investigate the cardinal number Add $(F_1,F_2)$ in the case when one of the families $F_1,\; F_2$ is Darboux-like or Sierpiński-Zygmund and the other one is the class of Hamel functions, where Add $(F_1,F_2)$ is defined as the smallest cardinality of a family $F \subseteq \mathbb{R}^{\mathbb{R}}$ for which there is no $g \in F_1$ such that $g + F \subseteq F_2$

We provide a simplification of Zahorski's argument showing that for every Lebesgue null $G_{\delta\sigma}$ subset $G$ of the line there is a Lipschitz function that is non-differentiable precisely at the points of $G$.

The motivation for this paper is due to a question from Z. Piotrowski on whether or not the "textquotedblleft salt-and-pepper" function in the plane was the pointwise limit of separately continuous functions. In this paper we answer that question and then go on to investigate the sets $D$ in the plane such that $\chi _{D}$ is the pointwise limit of separately continuous functions. We also look at all pointwise limits of separately continuous functions and their place in the space of Baire Class 2 functions.

In a little known paper, Haslam-Jones defined a collection of higher order derivatives in terms of an integral and Legendre polynomials. One member of this collection is equivalent to the higher order Peano derivatives. The purpose of this paper is to present a more direct proof of this equivalence.

In a 1990 paper by R. Mabry, it is shown that for any constant $a\in \left(0,1\right)$ there exist sets $A$ on the real line with the property that for any bounded interval $I$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}=a$, where $\mu$ is any Banach measure. \noindent Many of the constructed sets are Archimedean sets, which are sets that satisfy $A+t=A$ for densely many $t\in {\mathbb{R}}$. In that paper it is shown that if $A$ is an arbitrary Archimedean set, then for a fixed, $\mu$, $\displaystyle\frac{\mu(A\bigcap I)}{\mu(I)}$ is constant. (This constant is called the $\mu$-shade of $A$ and is denoted ${\rm sh}_{\mu}A$.) A problem is then proposed: For any Archimedean set $A$, any fixed Banach measure $\mu$, and any number $b$ between $0$ and ${\rm sh}_{\mu}A$, does there exist a subset $B$ of $A$ such that $\displaystyle\frac{\mu(B\bigcap I)}{\mu(I)}=b$ for any bounded interval $I$? In this paper, we partially answer this question. We also derive a lower bound formula for the $\mu$-shade of the difference set of an arbitrary Archimedean set. Finally, we generalize an intersection result from Mabry's original paper.

Some convergence theorems have been obtained for the $GH_k$ integral for functions defined in abstract topological spaces and with values in Riesz spaces.

The aim of the paper is to give sufficient and necessary conditions when the attractor of an iterated multifunction system, composed by multifunctions with compact and connected values, is a connected set.

There are derivations $f : \mathbb{R} \to \mathbb{R}$ which are almost continuous in the sense of Stallings but not extendable. Every derivation $f : \mathbb{R} \to \mathbb{R}$ can be expressed as the sum of two extendable derivations, as the discrete limit of a sequence of extendable derivations and as the limit of a transfinite sequence of extendable derivations. Analogous results hold for additive functions.

Let $\mu$ be a translation-invariant metric measure on $\mathbb{R}$ with the following scaling property: for every $\lambda \in (0,1)$ there exists $b(\lambda)>\lambda$ with $\mu(\lambda X)\geq b(\lambda) \mu(X)$ for all $X \subseteq \mathbb{R}$. If $X$ is a $\mathbb{Z}$-invariant subset of $\mathbb{R}$ with $X/q \subseteq X$ for some $q\in \mathbb{N}\setminus \{1\}$, then $\mu(X)=0$ or $\mu(X\cap O)=\infty$ for every non-empty open set $O$. This refines an earlier result by Olsen.

In this paper we give some equivalence relations between behavior of Fourier coefficients of a special kind and smoothness of functions. A necessary and sufficient condition for existence of Schwartz derivative is also obtained.

Although the sequence of greedy approximants associated with the Schauder expansion of a function, $f$, continuous on $[0,1]$, may fail to converge, there always will be a continuous function, arbitrarily close to $f$, whose Schauder expansion does have a convergent sequence of greedy approximants. Further examination of this problem shows that the same sort of proposition is valid for a multitude of subsystems of the Schauder system.

Using some properties of pseudocontinuous functions - a recent generalization of continuous functions - and without invoking Debreu's Open Gap Theorem, we solve the following problem: given a pseudocontinuous function $v$, find a continuous function $u$ such that $u(x)>u(y)$ if and only if $v(x)>v(y)$. We show that this problem can be solved only for pseudocontinuous functions. Finally, we obtain a new proof on the existence of continuous numerical representations for continuous, transitive and total binary relations.

Kostyrko and Šalát showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere. We give a topological analogue of this theorem and provide some examples.