In this paper we will consider nowhere dense perfect subsets of $[0,1]$ that arise as a natural generalization of symmetric perfect sets in a paper of Humke, called centered sets. These sets have relatively large basic intervals but none of the standard methods used in determining the dimension of a generalized Cantor set could be used to show that centered sets can not have zero Hausdorff dimension. The aim of this paper is to prove that and to give lower bound for this dimension..

We will consider the classes of first return continuous, weakly first return continuous with respect to some trajectory and almost continuous functions in the sense of Stallings and we will show that for the Baire $1$ functions the classes of functions mentioned above are equal. Moreover we will define the class $\mathcal{B}_{1}^{\mathcal{F}}$ of strongly $\mathcal{F}$-almost everywhere first return recoverable functions and shall present the relation between family $\mathcal{B}_{1}^{\mathcal{F}}$ and Baire $1$ functions.

Given a collection of functions of some class defined on the real line, when can you find a large set upon which the restriction of every function is continuous? We consider this problem (and related problems) for various classes of functions and various notions of largeness. These problems can be considered in terms of finding the covering, uniformity (non), additivity, and cofinality numbers for some ideal-like collections of sets.

Let $I$ be a compact real interval and $f\colon ~I\rightarrow I$ continuous. We describe a special infinite minimal subsystem - we call it irrational twist system - of dynamical system $(I,f)$. We show that any twist system has an extremely regular behavior and it can be considered as an interval analogy of the irrational circle rotation.

Let $I = [0,1]$, and let $Ext(I)$ or $Ext$ denote the subspace of all extendable connectivity functions $f:I \to {\mathbb R}$ with the metric of uniform convergence on $I^{\mathbb R}$. We show that $Ext$ is porous in the almost continuous function space $AC$ by showing that the space $AC \cap PR$ of all almost continuous functions with perfect roads is porous in $AC$. We also show that for $n >1$, the subspace $Ext({\mathbb R}^n)$ of all extendable connectivity functions $f:{\mathbb R}^n \to {\mathbb R}$ is a boundary set in the Darboux function space $D({\mathbb R}^n)$.

Let $f$ be a real (or complex) function with domain $D_f$ containing the positive integers. We introduce the functional sequence $\{ f_{\sigma _n}(x) \}$ as follows:$f_{\sigma _n}(x)=xf(n)+\sum_{k=1}^n(f(k)-f(x+k))$ and say that the function $f$ {\sl limit summable} at the point $x_0$ if the sequence $\{ f_{\sigma _n}(x_0) \}$ is convergent, $( f_{\sigma _n}(x_0)\rightarrow f_\sigma (x_0)) \; {\mbox as} \; n \rightarrow \infty$, and we call the function $f_\sigma(x)$ as the limit summand function (of $f$). In this article, we first give a necessary condition for the limit summability of functions and present some elementary properties. Then we prove some tests about limit summability of functions and consider the relation between $f(x)$ and $f_\sigma(x)$. One of the main theorems in this paper gives a uniqueness conditions for a function to be a limit summand function. Finally, as a consequence of this theorem we deduce a generalization of a result due to Bohr-Mollerup.

Some versions of Dieudonné theorems are given for set functions, not necessarily positive, taking values in Dedekind complete $(l) $-groups, relatively to the "$(D)$-convergence"

In this paper higher-order convexity properties of real functions are characterized in terms of a Dinghas-type derivative. The main tool used is a mean value inequality for Dinghas-type derivatives.

In this paper, we shall consider two generalized Riemann approaches to the Itô integral; namely, the Itô-Henstock and the Itô-McShane approaches, and by establishing the equivalence of the Itô-Henstock integral with the classical Itô integral, prove the equivalence of all the three integrals.

In this work we provide a characterization of $C^{k,1}$ functions on $\mathbb{r}^n$ (that is, $k$ times differentiable with locally Lipschitzian $k$-th derivatives) by means of $(k+1)$-th divided differences and Riemann derivatives. In particular we prove that the class of $C^{k,1}$ functions is equivalent to the class of functions with bounded $(k+1)$-th divided difference. From this result we deduce a Taylor's formula for this class of functions and a characterization through Riemann derivatives.

A set $E\subseteq \mathbb{R}^n $ is $s$-straight for $s>0$ if $E$ has finite Method II outer $s$-measure equal to its Method I outer $s$-measure. If $E$\ is Method II $s$-measurable this means $E$ has finite Hausdorff $s$-measure equal to its Hausdorff $s$-content. Here we make a first study of such sets, following their 1995 introduction by Foran. Primary facts are proved about subsets, intersections, unions, and some mappings of $s$-straight $s$-sets. Basic examples of $1$-straight and countable unions of $1$-straight $1$-sets are constructed from line segments. It is noted that self-similar $s $-sets are $s$-straight. Verifying a conjecture of Foran, the circle is proved to be the countable union of perfect $1$-straight $1$-sets along with a set of Hausdorff $1$-measure zero. Such perfect sets are then further examined. Also examined are subsets of $1$-straight sets $E$ maximal in the sense that their Hausdorff $1$-measure equals the diameter of $E$.

We generalize Bourgain's theorem on the two dimensional Kakeya maximal function by proving norm estimates with respect to measures satisfying certain conditions. We use this to extend the classical result of Davies on the Hausdorff dimension of Kakeya sets in the plane.

It is shown that a Lebesgue integrable function comes equipped with a sequence of points which one can use in conjunction with a simple ``first return -- Riemann'' integration procedure to compute the integral.

Let $(X,d,\mu)$ be a metric measure space of homogeneous type with a finite measure. Assume that $\Omega \subset X$ is a bounded domain, which satisfies an $A^*(\varepsilon,\delta)$-condition and $\mu(\partial\Omega)=0$. We show that there exists a bounded linear extension operator $Ext$ from the Hajłasz space $M^{1,p}(\Omega,d,\mu)$ to $M^{1,p}(X,d,\mu)$, such that $Ext (u)\vert_\Omega = u$.

In this article we investigate the maximal additive (multiplicative) [lattice] families for the class of real functions defined on topological spaces which are upper and lower semi-quasicontinuous at each point.

Recently the first author investigated a generalization of families of trigonometric thin sets replacing the sine function by a continuous function. In this paper we shall partially solve the problem of the relationship between such families obtained from different functions. In several cases we present conditions for equality of such a family with corresponding trigonometric family. Moreover we show that every basis of any of the families of $B_0$--sets, $N_0$--sets or $A$--sets has cardinality at least that of the continuum.

In this paper, give a new, full characterization of the primitive of a Kurzweil-Henstock integrable function in multidimensional space.

This paper is a step by step account of the Riesz approach to the Lebesgue integral. Besides, motivating the use of ``almost everywhere'' tools, we eliminate unnecessary equivalences, and we give a simple representation of a complete space of integrable functions, usually missing from classical treatises.

In this article we investigate some properties of functions $f:X\times Y \to {\cal R}$ (the cliquishness, the Baire property, the measurability) whose vertical sections $f_x$ have closed graphs.

In this paper, we give an example to show that a theorem of Nakanishi for the Henstock integral does not hold for the general Denjoy integral.

This paper discusses the decomposition of variational measures in $\mathbb{R}^{n}$ and, by using integral expressions of variational measure, gives an arc-length integral formula for the continuous curve in $\mathbb{R}^{n}$.

To characterize compact parts of spaces $L^p(\Omega)$, we introduce a concept of equi-integrability based on the approximation of elements of $L^p(\Omega)$ by simple functions. The resulting theorem will be used to develop a new methodology to prove and extend results about the compactness of Sobolev embeddings.

We show that (under CH) there exists a $CIVP$ function with a $\lambda^\prime$ graph. We examine some properties of $\mathcal{M}_\alpha$ and $\mathcal{M}_alpha(\lambda^\prime)$ class of real valued functions

In this paper we introduce a measure on an interval connected with variation of given function $f.$ Next we use this measure to calculate variation of a composition.

The Lebesgue and Bochner integrals are characterized by strong derivatives, inner variation and Lusin condition in this note.

We show that there exists a bi-Lipschitz homeomorphism $f\:RR\to \RR$ and a dense open set $U$ of the space of Lipschitz functions defined on $[0,1]$, for which $\{f\circ g\:, g\in U\}$ is of first category.

Jean Mawhin has proved a version of Stokes' theorem on a cube using a generalized Riemann integral. We give a new, much simpler, and intuitive proof of his theorem using the integral definition of the exterior derivative.

For an algebra $\mathcal{A}$ and an ideal $\mathcal{I}$ of subsets of a set $X$ we consider pairs $(\mathcal{A, I})$ which have the common inner Marczewski-Burstin representation. The main goal of the paper is to investigate which inner Marczewski-Burstin representable algebras and pairs are topological.

In classical works, Hurewicz and Menger introduced two diagonalization properties for sequences of open covers. Hurewicz found a combinatorial characterization of these notions in terms of continuous images. Recently, Scheepers has shown that these notions are particular cases in a large family of diagonalization schemas. One of the members of this family is weaker than the Hurewicz property and stronger than the Menger property, and it was left open whether it can be characterized combinatorially in terms of continuous images. We give a positive answer. We also find some additivity numbers for these classes. This paper can serve as an exposition of this fascinating subject.

We give characterizations of sets $E\subset[0,1]$ for which the local monotonicity of each function $f:[0,1]\to\mathbb{R}$ from a given class $\mathcal{F}$, at all points $x\in E$, implies the global monotonicity of $f$ on $[0,1]$. We consider as $\mathcal{F}$ -- the families of continuous functions, differentiable functions, absolutely continuous functions, functions of class $C^n$ ($n=1,2,...,\infty$), real analytic functions and polynomials.

In this paper the sets of discontinuity points and the sets of approximate discontinuity points of function $f:{\cal R} \to {\cal R},$ satisfying some special approximate quasi-continuity conditions introduced in \cite{2}, are investigated.

For any $C\subseteq \mathbb{R}$ there is a subset $A\subseteq C$ such that $A+A$ has inner measure zero and outer measure the same as $C+C$. Also, there is a subset $A$ of the Cantor middle third set such that $A+A$ is Bernstein in $[0,2]$. On the other hand there is a perfect set $C$ such that $C+C$ is an interval $I$ and there is no subset $A\subseteq C$ with $A+A$ Bernstein in $I$.

The almost disjointness number is extended to arbitrary Boolean algebras and it is shown that this number is consistently less than $\mathfrak a$ for the Boolean algebra $\mathcal {P} (\mathbb{N})/ \mathcal {N}$ where $\mathcal {N}$ is the ideal of nowhere dense subsets of $\mathbb{Q}.

We deal with two types of chaos: the well known chaos in the sense of Li and Yorke and $\omega$-chaos which was introduced by S. Li in 1993. In this paper we prove that every bitransitive map $f \in C(I,I)$ is conjugate to $g \in C(I,I)$, which satisfies the following conditions,

1. there is a $c$-dense $\omega$-scrambled set for $g$,

2. there is an extremely LY-scrambled set for $g$ with full Lebesgue measure,

3. every $\omega$-scrambled set of $g$ has zero Lebesgue measure.