During the past decade several authors have explored some ramifications of borrowing the ``first-return'' notion from the study of the dynamics of chaotic maps on an interval and applying the concept to the core areas of real analysis: approximation, continuity, differentiation, and integration. This survey attempts to summarize the results obtained and to point toward possible future directions for exploration.

Results are deduced from the following dichotomy which reduces descriptive properties concerning the Ellentuck topology to two well-known examples. Theorem:Every nonempty perfect set in the Ellentuck topology contains a closed copy of the Sorgenfrey line or a closed copy of the rational numbers. This leads to a Marczewski-Burstin representation for Marczewski sets in the Ellentuck topology.

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

Recent results concerning the convergence almost everywhere or divergence everywhere of Dirichlet series $\sum a_nn^{it}$ appeared in the literature, revealing significant differences with the case of trigonometric series $\sum a_n e^{int}$. In this work, we prove in several cases the optimality of these results. We also discuss the statistical effect of a change of signs, by considering $\sum \pm a_nn^{it}$. According to the way (probabilistic or topological) this change of signs is made, the properties of the resulting series are quite different, and can also be applied to the theory of power series.

We give three equivalent properties characterizing the essentially ejective sets of a compact commutative topological group.

In this paper we prove that if there is a Borel-cover $\gamma$-set of cardinality the continuum, then there is one which is not hereditary.

We study moment problems in which one searches for a probability density on ${\R}^n$ by using insufficient information given in an integral form. We characterize the existence of the representing densities for a finite multi--sequence of moments by the solvability of a concrete finite system of equations. Its solution provides the (unique) representing density of maximum entropy allowed by the given data, that turns out to be the exponential of a polynomial to be determined. For all densities of this form, the system to be solved can be taken linear to be provided sufficiently many moments are known.

In this paper we compute transitive cardinal coefficients of the $\sigma$-ideal $\mathbb{S}_2$, the least nontrivial productive $\sigma$-ideal of subsets of the Cantor space $2^\omega$. We also apply transitive operations to $\mathbb{S}_2$. In particular, we show that $\sigma$-ideal of strongly $\mathbb{S}_2$ sets is equal to $\mathbb{B}2$, one of Mycielski ideals.

WWe prove the following extension of the Mean Value Theorem. Let E be a Banach space and let $F:[a,b]\rightarrow E$and $\varphi :[a,b]\rightarrow\mathbb{R}$ be two functions for which there exists a subset $A\subset\lbrack a,b]$ such that:

i) $F$ and $\varphi$ have negligible variation on $A$,

ii) $F$ and $\varphi$ are differentiable on $[a,b]\setminus A$ and $\left\Vert F^{\prime}\right\Vert \leq\varphi^{\prime}$ on $[a,b]\setminus A$.

Then $\left\Vert F(b)-F(a)\right\Vert \leq\varphi(b)-\varphi(a).$

Several applications are included.

In the one-dimensional case it is well-known that functions of bounded variation on $\mathbb{R}$ possess at most a countable number of non-regular points. In this paper we will show that multivariate functions $f:\mathbb{R}^n \rightarrow \mathbb{R} $ of bounded variation satisfying the condition lim$_{|x| \to \infty} f(X) $ are non-regular at most on a subset of $\mathbb{R}^n$ of Lebesgue measure zero. Moreover, we will point out that this result is best possible.

We prove: In the space ${\mathcal C}$ of continuous functions on $[0,1]$ under the $sup$ metric, the functions all of whose level sets (in every direction) have measure zero, form a residual subset of ${\mathcal C}$. In the space ${\mathcal D}$ of bounded derivatives of $[0,1]$, the derivatives all of whose level sets are nowhere dense sets of measure zero form a residual subset of ${\mathcal D}$. Moreover, there exists a derivative in ${\mathcal D}$ all of whose level sets have measure zero and one of whose level sets is dense in $[0,1]$.

The goal of this paper is to describe the panorama of Mathematics grown up from the celebrated inequality of Hermite and Hadamard. Both old and new results are presented, complemented and discussed within this framework.

The graph of a function $f$ is subjected to non-homogeneous dilatations around the point $(x_0;f(x_0))$, related to the Taylor's expansion of $f$ at $x_0$. Some questions about convergence are considered. In particular the dilated images of the graph are proved to behave nicely with respect to a certain varifold-like convergence. Further and stronger results are shown to hold in such a context, by suitably reinforcing the assumptions.

Let $f: X \rightarrow {\mathbb R}$, where $X$ is a subset of ${\mathbb R}^{k}$. We observe that in order for $f$ be almost everywhere pointwise Hölder it is enough that $f$ satisfy the Hölder condition inside angles of a fixed width. In this analysis, density points of $X$ play a primary role. This has some interesting consequences concerning summability of a naturally defined coefficient.

On a separable Banach space, we show that a cone-monotone function is generically intermediate differentiable provided its Dini-derivatives are finite along every direction and the cone has nonempty interior.

We consider an infinite game on a group $G$, defined relative to a subset $A$ of $G$. The game is denoted $\mathsf{G}(G,A)$. The finite version of the game, introduced in [1], was inspired by an attack on the RSA crypto-system as used in an implementation of SSL. Besides identifying circumstances under which player TWO does not have a winning strategy, we show for the topological group of real numbers that if $C$ is a set of real numbers having a selection property (*) introduced by Gerlits and Nagy, then for any interval $J$ of positive length, TWO has a winning strategy in the game $\mathsf{G}(\mathbb{R},J \cup C)$

We prove that the Covering Property Axiom $CPA_{prism}, which holds in the iterated perfect set model, implies the following facts:

There exists a family $\mathcal{G}$ of uniformly continuous functions from $\mathbb{R}$ to $[0,1]$ such that $|\mathcal{G}|=\omega_1$ and for every $S\in[\mathbb{R}]^\mathfrak{c}$ there exists a $g\in\mathcal{G}$ with $g[S]=[0,1]$

The additivity of the Marczewski's ideal $s_0$ is equal to $\omega_1<\mathfrak{c}$.

The McShane integral of Banach space-valued functions $f:I\to X$ defined on an $m$-dimensional interval $I$ is considered in this paper. We show that a McShane integrable function is integrable over measurable sets contained in $I$ (Theorem 9). A certain type of absolute continuity of the indefinite McShane integral with respect to Lebesgue measure is derived (Theorem 11) and we show that the indefinite McShane integral is countably additive (Theorem 16). Allowing more general partitions using measurable sets instead of intervals another McShane type integral is defined and we show that it is equivalent to the original McShane integral (Theorem 21)

We prove that if $f$ is the maximum of two unilaterally continuous regulated functions, then the set $D_{un}(f)=\{x:f$ is not unilaterally continuous at $x\}$ is unilaterally isolated and for $x \in D_{un}(f)$ the inequality $f(x)<\max(f(x+),f(x-))$ holds. Moreover, for a regulated function $f$ such that $D_{un}(f)$ is isolated and for $x \in D_{un}(f)$ the inequality $f(x)<\max(f(x+),f(x-))$ holds, there are two unilaterally continuous regulated functions $g$, $h$ with $f=\max(g,h)$.

In this note we consider two natural attempts to give a descriptive characterization for $H_1$-primitives, and discuss why these attempts fail. Meanwhile we get a new descriptive definition of the Henstock integral. Also, we prove that every Henstock integrable function can be written as a sum of a Lebesgue integrable and an $H_1$-integrable ones.

We define and investigate the symmetric derivative for functions defined on a subset of the real line. We give an example of a continuous function with a positive symmetric derivative everywhere which is not monotonic. When the domain is measurable or has the Baire property, then a positive symmetric derivative does imply monotonicity on a big set.

Denote by $\mu$ a probability Borel measure on the real line and by $\tau_c$ the translation by $c.$ We show that $\mu$ is singular with respect to Lebesgue measure if and only if the set of those $c$ for which $\mu$ and $\tau_c\mu$ are mutually singular is dense (Theorem 1). Another characterization of singularity (Theorem 10) is the existence of a set of full $\mu$ measure that has continuum many disjoint translates. This result is also linked to some known results about $\sigma$-porous sets

The purpose of this note is to show that if $-\infty < \alpha < \beta < \infty $ and $E_{\alpha }^{\beta }$ is the equivalence relation, which is defined on the Polish group $C([\alpha ,\beta ), {\mathbb R}_{+}^{*})$ by $fE_{\alpha }^{\beta }g \iff \lim_{x\rightarrow {\beta }^{-}}\frac{f(x)}{g(x)}=1$, where $f$, $g$ are in $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$, then $E_{\alpha }^{\beta }$ is induced by a turbulent Polish group action. Hence if $L$ is any countable language and ${\mathcal{A}}:C([\alpha ,\beta ),{\mathbb R}_{+}^{*})\rightarrow X_{L}$ is any Baire measurable function from the Polish group $C([\alpha ,\beta ), {\mathbb R}_{+}^{*})$ to the Polish space $X_{L}$ of countably infinite structures for $L$ with the property that $fE_{\alpha }^{\beta }g \Rightarrow {\mathcal{A}}(f) \cong {\mathcal{A}}(g)$, whenever $f$, $g$ are in $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$, then there exists a $E_{\alpha }^{\beta }$-invariant comeager subset $S$ of $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$ for which all countable structures in ${\mathcal{A}}[S]$ are isomorphic.

An algorithm is given for constructing polynomials that determine approximately whether a point $p$ is inside or outside a given polyhedron $C_n$ in Euclidean $n$-dimensional space. The polynomials are of degree $2r$, where $r$ is a positive integer and the order of the approximation can be made arbitrarily small by taking $r$ sufficiently large. For $n=2$, the square, triangle, trapezoid, and pentagon are used as examples. For $n=3$ and $n=4$, the tetrahedron and equilateral simplex are used as examples. We conjecture that the center of mass of the region determined by the approximating polynomial is the same for all values of $r$, and hence coincides with the center of the polyhedra.

Let $A(x)$, $B(x)$, $C(x)$ be characteristic functions of three measurable sets of real numbers. We determine necessary and sufficient conditions for which $A(x+a_{n})+B(x+b_{n})+C(x+c_{n})=A(x)+B(x)+C(x)$ almost everywhere, where $\{ a_n\},\{ b_n\},\{ c_n\}$ are sequences of nonzero shifts approaching zero.

In his lecture [1] Borsík introduces some notions of the bilateral quasicontinuity of applications of topological spaces. In this paper I define analogously the bilateral semi-quasicontinuities and investigate some functions of two variables whose sections are bilaterally quasicontinuous or bilaterally semiquasicontinuous.

We give a characterization of the quantization dimension of Borel probability measures on $\mathbb{R}^{d}$ in terms of $\epsilon$-quantization numbers. Using this concept, we show that the upper rate distortion dimension is not greater than the upper quantization dimension of order one. We also prove that the upper quantization dimension of a product measure is not greater than the sum of that of its marginals. Finally, we introduce the notion of the $\epsilon$-essential radius for a given measure to construct an upper bound for its quantization dimension.

Let $1<p\leq q<\infty .$ Inspired by some recent results concerning Hardy type inequalities we state and prove directly the equivalence of four scales of integral conditions. By applying our result to the original Hardy type inequality situation we obtain a new proof of a number of characterizations of the Hardy inequality and obtain also some new weight characterizations. As another application we prove some new weight characterizations for embeddings between some Lorentz spaces.

We introduce and discuss a condition generalizing one of the Archimedean properties characterizing parabolas.

On any uncountable Polish space we construct a sequence of continuous functions $(f_n)$ such that $\sum f_{n}$ is divergent everywhere, but for a typical sign sequence $(\varepsilon_n) \in \{-1, +1\}^{\mathbb{N}}$, the series $\sum \varepsilon_{n} f_{n}$ is convergent in at least one point. This answers a question of S. Konyagin in the negative.

In this paper we prove that if $X$ is an uncountable subset of the reals and $\kappa $ is a cardinal smaller than the cardinality of the set $X $ then the algebraic difference $X-X$ of the set $X$ is not a finite union of $\kappa $ sum free or $\kappa$ difference free sets. An application of the above result is that for any function $f:\mathbb{R}\to \{1,2,...,n\} $ and for each cardinal $\lambda <2^{\omega } $the set of all $x $ such that ${|\{h>0:f(x-h)=f(x+h)\}|\geq \lambda } $ is of the size of the continuum. Among other things, we show that a finite union of countably many translates of $2^{\omega } $ difference free subsets of the reals is not residual in an interval. In the above statement, \textquotedblleft countably many\textquotedblright can be replaced by "fewer than continuum many" provided that $2^{\omega } $ is a regular cardinal

We present conditions under which one may substitute the identity function for $h$ in Kurzweil-Henstock integrals of the form $\int(f\circ h)\,d(g\circ h)$ reducing them to equivalent integrals of the form $\int f\,dg$\,. Our study requires that we also consider reduction of $\int (f\circ g) |d(g\circ h)|$ to $\int Nf|dg|$ where $N$ is the Banach indicatrix of $h$.

Let $X$ be an interval, $Y$ a metric space, $T$ a set of parameters, and $f:T\times X\to Y$ a function. For given $t\in T$ denote by $v(t)$ the total variation of $f(t,\cdot )$ on $X$. We look for sufficient conditions for regular (measurable, continuous, etc.) dependence of $v$ on $t$.

We show that if $f,g\colon[0,1]\to[0,1]$ are both functions with connected $G_\delta$ graphs, then their composition has a fixed point. This is a generalization of the analogous result for Darboux Baire~1 functions.

In this paper the relation between N. Wiener's theorem about characterization of irregular points for the Dirichlet problem and the uniqueness problem for Fourier series is established.

A complete version of Lebesgue's differentiation theorem, including the image of the exceptional set, is proved in an elementary way.

A short proof is given for the Lebesgue Differentiation Theorem using a variation of the Heine-Borel covering property, without reliance on sophisticated approaches such as Vitali covers and the rising sun lemma.

The standard techniques used to prove the Lebesgue differentiation theorem (that monotonic functions are a.e.~differentiable) are presented in an unusual way that reveals more about their nature and allows greater generality.

We show various generalizations of the theorem from [3] concerning MB-representation of the interval algebra on $[0,1)$

The aim of this note is to show that for a given continuous function $F$ on a set $E \subset \R,$ the associated interval function need not be continuous. We also give an example to show that the associated interval function can be continuous even if $F$ is not continuous.