This note corrects an oversight in the paper MB-representations and topological algebras, Real Analysis Exchange 27(2), pp. 749-765.

My aim in this article is to provide an accessible introduction to the notion of topological entropy and (for context) its measure theoretic analogue, and then to present some recent work applying related ideas to the structure of iterated preimages for a continuous (in general non-invertible) map of a compact metric space to itself. These ideas will be illustrated by two classes of examples, from circle maps and symbolic dynamics. My focus is on motivating and explaining definitions; most results are stated with at most a sketch of the proof. The informed reader will recognize imagery from Bowen's exposition of topological entropy [Bow78] which I have freely adopted for motivation.

Using the notions of uniform equal and uniform discrete convergence for sequences of real valued functions, the classes of functions which are uniform equal limits and uniform discrete limits of sequences of real valued functions belonging to certain class are studied. Also, new types of convergence of sequences of real valued functions, called $\alpha$-uniform equal, $\alpha$-strong uniform equal and $\alpha$-equal are defined and studied. Using $\alpha$-uniform equal convergence, a characterization of compact metric space is obtained.

The present paper concerns a general version of the Henstock integral that includes Schwabik's generalized Perron integral and all the Stieltjes type integrals induced by measures as well.

Let $f$ be an element of $C(I,I)$ with $R(f)=\{x\in I:x\in \omega (x,f)\}$ its recurrent set. We study the relationship between the structure of $R(f)$ and the chaotic nature of the function $f$ . We show that $R(f)$ is always a $G_{\delta }$ set whenever $f$ has zero topological entropy, although $R(f)$ is closed for the typical continuous function $f$ with zero topological entropy. We also develop necessary and sufficient conditions on $f$ for $R(f)$ to be closed.

Let $0<a<\sqrt 2$. Suppose $\delta=\delta(d,\eps)$ has the following property. If $\mathcal N$ is an $a$-net of the Euclidean ball in $\RR^{d}$, $A\subset \mathcal N$, and $f:A\to \RR^d$ is $(1+\eps)$-bilipschitz, then $f$ admits a $(1+\delta)$-bilipschitz extension $f:\mathcal N\to \RR^d$. We give some estimates of $\delta$.

The aim of the paper is to characterize the class of sets of points at which a sequence of real functions of a distinguish family $\mathcal{F}\subset \mathbb{R}^X$ discretely converges.

A function $f:{\R } \to {\R }$ is strongly approximately quasicontinuous at a point $x$ if for each real $r > 0$ and for each set $U \ni x$ belonging to the density topology there is an open interval $I$ such that $I \cap U \neq \emptyset $ and $f(U\cap I) \subset (f(x)-r,f(x)+r)$. In this article we investigate the sets $A$ such that each almost everywhere continuous bounded function may be extended from $A$ to a bounded strongly approximately quasicontinuous function on ${\R }$.

We will prove a certain characterization of the function $x^2$ and of some similar functions, in the style of Cauchy's characterization of the function $ax$ by its additivity and boundedness over any interval of positive length.

We prove that for any $a, c \in (0,1)$ and any $b,d \in {\mathbb R},$ the Hausdorff dimension of $ \{ x\in [0,1] : n^{-a} \sum_{j=1}^{n}r_j(x) \to b \hbox{ and } n^{-c} \sum_{j=1}^{n} r_j(x) r_{j+1}(x)\to d\},$ is equal to 1, where $\{r_n(x)\}_{n \geq 1},$ are the Rademacher functions. We give also an extension of this result.

We construct a family of Dirichlet sets of cardinality $\mathfrak c$ such that the arithmetic sum of any two members of the family contains an open interval. As a corollary we obtain that every basis of many families of thin sets has cardinality at least $\mathfrak c$. Especially, every basis of any of trigonometric families $\mathcal{D}$, $\mathcal{pD}$, $\mathcal{B}_0$, $\mathcal{N}_0$, $\mathcal{B}$, $\mathcal{N}$, $\mathcal{wD}$ and $\mathcal{A}$ has cardinality at least $\mathfrak c$. Moreover, we construct an increasing tower of pseudo Dirichlet sets of cardinality $\mathfrak t$.

A function $f:{\mathR} \to {\mathR}$ satisfies condition $(s_3)$ if for each real $\varepsilon > 0$, for each $x$ and for each set $U \ni x$ belonging to the density topology there is an open interval $I$ such that $A(f) \supset I \cap U \neq \emptyset $ and $f(U\cap I) \subset (f(x)-\varepsilon ,f(x)+\varepsilon )$, where $A(f)$ denotes the set of all approximate continuity points of $f$. In this article it is show that the sum of two functions with the condition $(s_3)$ is the sum of two Darboux functions satisfying this condition $(s_3)$ and that every a.e.-continuous function with some special condition is the sum of two functions with condition $(s_3).$

In this paper we continue the research in Real Analysis, in the spirit of the very recent book [1]. Firstly, as a refinement of the global defect of integrability introduced in [1], $\S $5.1, we consider here a pointwise defect of integrability and study its properties and connections with the pointwise defect of continuity already introduced in [1], $\S$ 5.1. Secondly, as refinements of the global defects of monotonicity and of convexity introduced in the same book [1], $\S$ 5.2, we consider and study pointwise variants

Necessary and sufficient for $\int_a^bfg_n\to \int_a^bfg$ for all Henstock--Kurzweil integrable functions $f$ is that $g$ be of bounded variation, $g_n$ be uniformly bounded and of uniform bounded variation and, on each compact interval in $(a,b)$, $g_n\to g$ in measure or in the $L^1$ norm. The same conditions are necessary and sufficient for $\|f(g_n-g)\|\to 0$ for all Henstock--Kurzweil integrable functions $f$. If $g_n\to g$ a.e., then convergence $\|fg_n\|\to\|fg\|$ for all Henstock--Kurzweil integrable functions $f$ is equivalent to $\|f(g_n-g)\|\to 0$. This extends a theorem due to Lee Peng-Yee.

WWe show that the existence of a loop of covering intervals for connected $G_\delta$ real functions implies the existence of a periodic point with its period equal to the length of the cycle. As a corollary we get that the composition of $N$ connected $G_\delta$ real functions from a compact interval into itself has a fixed point for every natural number $N$. In particular, the composition of finitely many derivatives from $[0,1]$ to $[0,1]$ has a fixed point. This solve the a problem from [3].

In this paper we look at various generalizations of continuity for a function and determine necessary additional conditions which result in continuity in the ordinary sense.

The main results of the paper, answering an open problem raised in [3], show that $t$-convexity can also be characterized in terms of a lower second-order generalized derivative. As a consequence, we obtain that $t$-convexity is also a localizable convexity property.

Let $M:(0,\infty)^{2}\rightarrow(0,\infty)$ be a homogeneous strict mean such that the function $h:=M(\cdot,1)$ is twice differentiable and $0\neq h^{\prime}(1)\neq1$. It is shown that if there exists an $M$-affine function, continuous at a point which is neither constant nor linear, then $M$ must be a weighted power mean. Moreover the homogeneity condition of $M$ can be replaced by $M$-convexity of two suitably chosen linear functions. With the aid of iteration groups, some generalizations characterizing the weighted quasi-arithmetic means are given. A geometrical aspect of these results is discussed.

Topological properties of family path continuous functions are investigated. We consider weak path continuity, path limit set and properties of $\sigma$-systems in metric spaces.

A stronger form of the Arzelá convergence is defined and it is compared to other types of convergence.

We investigate algebras of sets, and pairs $(\mathcal{A , I})$ consisting of an algebra $\mathcal{A}$ and an ideal $\mathcal{I} \subset \mathcal{A}$, that possess an inner MB-representation. We compare inner MB-representability of $(\mathcal{A , I})$ with several properties of $(\mathcal{A , I})$ considered by Baldwin. We show that $\mathcal{A}$ is inner MB-representable if and only if $\mathcal{A} =S(\mathcal{A} \setminus\mathcal{H}(\mathcal{A}))$, where $S(\cdot)$ is a Marczewski operation defined below and $\mathcal H$ consists of sets that are hereditarily in $\mathcal{A}$. We study the question of uniqueness of the ideal in that representation..

The Riemann approach to integration is well-known for its explicitness and directness. In this paper we use the Non-Uniform Riemann Approach to give an alternative definition of the Multiple Itô-Wiener Integral and prove that our definition is in fact equivalent to the classical definition.

We show that if $(X, \|. \|)$ is a separable Banach space, $\Omega \subset X$ is open, connected and $f: \Omega \to \mathbb{R}$ is an everywhere Gâteaux differentiable Lipschitz continuous function, then the graph of the derivative of $f$ is connected in $(\Omega,\| . \|)\times(X^\star , w^\star)$.

The crucial concept for studying quasi-continuous and cliquish functions on arbitrary topological spaces $X$ is the concept of a semi-open subset of $X$. On the one hand, it gives rise to the cluster set $SO-C(f;x)$ of a function $f:X\rightarrow {\mathbb R}$ at a point $x\in X$, which turns out to be an appropriate tool for investigating both local and global properties of $f$. On the other hand, the concept of a semi-open set is used for introducing so-called semi-open partitions of $X$. A central result of the paper says that every quasi-continuous function can be represented as a uniform limit of step functions defined on a chain of semi-open partitions of $X$. Similarly, every cliquish function is proved to be the uniform limit of step functions defined on a chain of so-called almost semi-open partitions of $X$.

The paper is related to the following question of P.L.Ul'yanov. Is it true that for any $2\pi$-periodic continuous function $f$ there is a uniformly convergent rearrangement of its trigonometric Fourier series? In particular, we give an affirmative answer if the absolute values of Fourier coefficients of $f$ decrease. Also, we study how to choose $m$ terms of a trigonometric polynomial of degree $n$ to make the uniform norm of their sum as small as possible.

We prove that the Lebesgue differentiation theorem holds in the general setting of spaces of homogeneous type if the balls are subspaces of homogeneous type.

The Dvoretzky covering problem is to cover the circle with random intervals. We consider the covering of the circle with random open sets. We find a necessary and sufficient condition for the circle to be covered almost surely when each open set is composed of a finite number of intervals which are separated by a positive distance.

V. Jarnik showed that a typical continuous function on the unit interval $[0,1]$ has every extended real number as a derived number at every point of $[0, 1]$. In this paper we classify the derived numbers of a continuous function and study the likelihood of Jarnik's Theorem for path derived numbers of a continuous system of paths. We also provide some results indicating that some of the nice behaviors of first return derivatives are shared by path derivatives of continuous functions when the path system is continuous.

We show how a combination of three different second derivatives can yield convexity, even though individually, none of them is sufficient.

In this article we investigate approximate semicontinuity of a function related to Grande's SAC problem.

In 1992, Šikić gives a characterization of Riemann-integrable functions as uniform limits of simple functions. The aim of this article is to prove an extension to the case of functions defined on a basic space $(X,{\cal D},\mu)$ and with values in any Banach space$F$

A new simple proof of the Ergodic Maximal Equality is proposed.

We answer a question which was stated by R. E. Svetic in [11]. The Bergelson-Hindman-Weiss lemma, which was placed in [1], is improved.

Some versions of Henstock-Kurzweil integral with respect to different derivation bases for functions with values in Dedekind complete Riesz spaces are studied. Fundamental Theorem of Calculus are proved for these integrals and an application to Walsh series is given.

Some examples, due to G. Birkhoff, are used to explore the differences and peculiarities of the Henstock and Kurzweil integrals in abstract spaces. We also include a proof, due to C. S. Hönig, of the fact that the Bochner-Lebesgue integral is equivalent to the variational Henstock-McShane integral.

It is shown that, similarly to the median, percentiles of a random variable can be characterized as minima of suitable functionals.

In this note we give an elementary proof that an arbitrary convex function can be uniformly approximated by a convex \cinf-function on any closed bounded subinterval of the domain. An interesting byproduct of our proof is a global equation for a polygonal (piecewise affine) function.

We prove that if $(\Omega,\Sigma,\mathcal{I}),\; (\Theta,T,\mathcal{J})$ and $(\Omega \times\Theta,\Xi,\mathcal{K})$ are measurable spaces with $\sigma$-ideals satisfying some natural Fubini type conditions then there is no density on $(\Omega\times\Theta,\Xi,\mathcal{K})$ with density invariant sections.

We generalize the functionally-small-Riemann-sum (FSRS) property for the Henstock integral to the $n$-dimensional Euclidean space. We prove a convergence theorem and its connection with the equi-integrability condition.

In this paper we shall show that the set ${\cal C}$ of all bounded continuous functions is superporous in the space ${\cal DB}_{1}^{**}$. Moreover, for an arbitrary function $f$ defined on ${\cal C}$ there exists a quasi-continuous extension $f_{1}$ of this function on ${\cal DB}_{1}^{**}$, such that ${\cal C}$ is the set of all discontinuity points of $f_{1}$.

Denote by $D_{\gamma}(P)$ the set of those points where the lower Lebesgue density of $P\subset \mathbb{R}$ is bigger or equal than $\gamma.$ We show that if $\gamma>0.5$ then $D_{\gamma}(P)\cap P$ is always of first category in any nowhere dense perfect set $P$. On the other hand, there exists a fat Cantor set $Q$ which is a subset of $D_{0.5}(Q)$ while for other fat Cantor sets $P$ it is possible that $D_{+}(P)=\cup_{\gamma>0}D_{\gamma}(P)$ is of first category in $Q.$