Over the last decade considerable progress has been made in developing extensions of the Lebesgue integral which provide the Gauss-Green theorem for noncontinuously differentiable vector fields and remain invariant under groups of transformations including diffeomorphisms. One particular extension, an averaging process defined by W.F. Pfeffer, accomplishes this in the setting of bounded sets of bounded variation—the most general class of sets for which the notions of “surface area” and “exterior normal” can be profitably defined. In this survey article we recover all of the notable geometric features behind Pfeffer’s extension in the less forbidding (vs. bounded sets of bounded variation) setting of figures, i.e. finite unions of compact intervals.

This paper summarizes the results related to the study of the continuity of multifunctions, separate and joint continuity, selection theorems, limits of multifunctions and generalized derivatives. The concept is based on the notion of Baire continuity and its connection to the topics mentioned above is given.

This paper describes the relationship between the behavior of continuous functions and their intersections with lines and other families of functions. The behavior is given in terms of differentiability, monotonicity and bounded variation. We see how arbitrary continuous function behave with respect to intersection patterns and we see that restricting the intersections to small sets gives certain desirable properties for the function.

For , let ${\text{Ω}}_{\alpha }={\displaystyle \prod \left[2^{-\alpha }-a_i,2^{-\alpha }+a_i\right]}$ be the probability space of Kahane and Salem, where $\left\{a_i\right\}$ is a positive sequence satisfying certain decaying condition. Let ${\left\{R_i\right\}}_1^{\infty }$ be the sequence of Rademacher functions. We show that for each $\alpha \in (0,1)$ and for almost all $\left\{{\mathrm{\xi }}_i\right\}\in {\mathrm{\Omega }}_{\alpha }$, the graph of $f(x)={\displaystyle \sum _{i=1}^{\infty }{\mathrm{\xi }}_1\cdots {\mathrm{\xi }}_i{R}_i(x)}$ has Hausdorff dimension $2-\alpha$.

In [4] Z. Grande asked, “What is the smallest lattice of functions containing all derivatives?” In this paper I prove that the answer is the family of all Baire one functions and that this family is the smallest lattice of functions containing all non-degenerate derivatives. Moreover it is proved that the lattice generated by bounded (non-degenerate) derivatives is the family of all bounded Baire one functions.

There are many ways to characterize a set $A$ as “thick” or “thin”. Discussed is an alternate way to provide such a characterization, via the difference set of $A$. Further, we discuss properties of the difference sets of decompositions $C$, $\tilde{C}$ of $ℝ$. We discuss in detail the possibilities that can arise when one encounters difference sets of Bernstein sets.

In connection with the study of $A{C}^{*}G$ functions, Lee Peng Yee introduced a condition which lies somewhere between $AC$ and Lusin’s condition $(\text{N})$, and it is called the strong Lusin condition. This condition also appears in Bordaon’s Lemma 2 of [4]. (In [7] Lee and Vyborny mentioned that this condition was also studied by Kurzweil, Jarnik and Schwabik.) Denoting this condition by $Y_{D^0}$, we show that $Y_{D^0}=A{C}^{*}G\cap \mathcal{C}$ on a closed interval.

There are also given several characterizations for the classes ${\underset{¯}{AC}}^{*}G\cap {\mathcal{C}}_i$, $AC$ and $\underset{¯}{AC}$. For these tasks we have developed a study of various interesting conditions, such as: $VB$, $\underset{¯}{VB}$, $V{B}^{*}$, ${\underset{¯}{VB}}^{*}$, $AC$, $\underset{¯}{AC}$, $A{C}^{*}$, ${\underset{¯}{AC}}^{*}$, $A{C}^{**}$, ${\underset{¯}{AC}}^{**}$, $A{C}_{D^{\#}}$, ${\underset{¯}{AC}}_{D^{\#}}$, $A{C}_{D^0}$, ${\underset{¯}{AC}}_{D^0}$, $A{C}_D$, ${\underset{¯}{AC}}_D$, $Y_{D^{\#}}$, ${\underset{¯}{Y}}_{D^{\#}}$, $Y_{D^0}$, ${\underset{¯}{Y}}_{D^0}$, $Y_D$, ${\underset{¯}{Y}}_D$ ($A{C}_{D^0}$ and $A{C}_D$ were introduced by Gordon in [4]).

A new characterization of the class of Baire 1 functions is given in terms of the notion of first return recoverability.

It is proven that every bounded upper semicontinuous function is the sum of two bounded upper semicontinuous and bilaterally quasicontinuous functions.

The set of all bilateral quasicontinuty points of real functions is investigated.

An improved version of the classical Bôcher’s theorem in $ℂ$ is proved using Carathéodory’s inequality and Riesz integral representation of subharmonic functions.

We reprove two theorems stated in the paper [6]. We propose a proof that does not use the regularity of the cardinal number continuum.

We reprove two theorems stated in the paper [6]. We propose a proof that does not use the regularity of the cardinal number continuum.

The paper contains several useful definitions which lead to the following monotonicity theorem: if $F$ is $C_d\cap (\overline{M})$ on $[a,b]$ and $F^{\prime }(x)\le 0$ a.e. where $F$ is derivable, then $F$ is decreasing on $[a,b]$.

By replacing the $\sigma$-ideal of meager sets with an arbitrary $\mathcal{I}$ ideal of codense sets, a theorem of $\text{M}$. Wilhelm is generalized. In particular, it is shown that every function $f:X\to Y$ that is almost continuous (with respect to $\mathcal{I}$) and which has the property of Baire (with respect to $\mathcal{I}$) is $\theta$-continuous.

Let $f:X\to Y$ be a function such that $f(X)=Y$ and let $F_f(y)=f^{-1}(y)$ for $y\in Y$. Some properties of the multifunction $F_f$ are investigated.

If a cyclic permutation which is not a double has a representative in a piecewise weakly monotone map of an interval, then it has a fundamental representative in that map.

In this note it is proved that the range of uniformly antisymmetric function must have at least four elements. This generalizes the results from [3] and [2] that the range of such function must have at least three elements. The problem whether the range of uniformly antisymmetric function can be finite remains open.

Let $\mathrm{\lambda }(A)$ denote the Lebesgue measure of $\text{A}$ and let $\mathrm{△}(A,I)$ denote the relative measure $\lambda (A\cap I)/\lambda (I)$.

We construct two disjoint open sets $A,B\subset [0,1]$ each having measure $1/2$ such that ${\lim }_{h\to 0}△(A,(x,x+h))-△(A,(x-h,x))=0$ (in fact uniformly in $x$). This says that the cumulative measure function $F(x)=\lambda (A\cap (0,x))$ is “uniformly smooth”, meaning that $F(x+h)-2F(x)+F(x-h)=o(h)$ (uniformly in $x$).

In this paper we obtain an estimate for the $k$-dimensional Hausdorff measure of a set on the boundary of a domain $D$ in ${ℝ}^n$ intersecting each $(n-k)$-dimensional linear subspace in at least 2 points. To do this we prove a version of Crofton formula, which allows us to reduce the problem to the question about the existence of tangent planes and derivatives. In general, a Borel set may not admit tangent planes at any of its points. We introduce the notion of density tangent planes and prove that every set with the finite $k$-dimensional Hausdorff measure has density tangent planes at almost all of its points. We compute derivatives of some set functions at such points and then use the Crofton formula to get the final result.

It follows from the obtained estimate that the $k$-dimensional Hausdorff measure of such a set is not less than the minimal area of sections of $\partial D$ by $k+1$-dimensional linear subspaces when $D$ is a ball or a cube. This purely real problem has a complex application. The $1$-dimensional Hausdorff measure of a set on the boundary of the unit ball or the unit cube in ${ℂ}^n$ with its polynomial hull containing the origin is not less than $2\pi$ or $4$, respectively.

In this paper we present a proof of SH. T. Tetunashvili that shows that a multiple trigonometric series that converges rectangularly everywhere actually converges iteratively everywhere to the same function. This method then solves a uniqueness problem, namely, that if a multiple trigonometric series converges rectangularly everywhere to zero, then all the coefficients are zero. We give a detailed proof in two dimensions. The result for higher dimensions may then be obtained inductively using the same proof.

This paper deals with the question of finding sets of functions which are determined by their values on dense sets. The main result gives a new condition sufficient for a class of functions to have that property. The condition implies the known facts that approximately differentiable functions and separately continuous functions are determined by dense sets. Also presented is the result that approximately differentiable functions on the unit interval are quasi-continuous. A lemma is given showing that if $X$ is a topological space, and $D$ is a dense semi-closed subset of $X$, then $D$ is the whole space $X$.

We present a potential theoretic proof of the following inequality of C. D. Cutler and L. Olsen [C. D. Cutler & L. Olsen, A Variational Principle for the Hausdorff Dimension of Fractal Sets, Math. Scand.(to appear)]: If $E$ is a Borel subset of ${ℝ}^d$, then

$$\dim E\le \underset{\mu \in 𝒫(E)}{\sup }\underset{¯}{R}(\mu )$$

where dim $E$ denotes the Hausdorff dimension of $E$, $𝒫(E)$ denotes the family of Borel probability measures supported by $E$, and $\underset{\_}{R}(\mu )$ denotes the lower Rényi dimension of the measure $\mu$.