If a function $F(x)$ is first-return differentiable to $f(x)$ then it is also universally first-return differentiable to $f(x).$

A construction is put forward which shows how to decompose an arbitrary continuous function $f\in C[0,1]$ into a sum of two continuous functions each having a graph of Hausdorff dimension one. An example is given where the terms in the decomposition have graphs of Hausdorff dimension 1 and packing dimension 2.

Let $\{\alpha_{n}\}_{n=0}^{\infty } $ denote a sequence of positive real numbers and let the sequence $\{\beta_{n}\}_{n=0}^{\infty } $ be defined by $\beta_{0} = 1$ and $\beta_{n+1} = \prod_{j=0}^{n}\alpha_{j}$. For $0 \leq a <1$, $0 < t < 1$, and $n$, a nonnegative integer, the inequality $0 \leq \{\xi \beta_{n} - a\} \leq t$ is studied, where $\{x\}$ denotes the fractional part of $x$. Let $\delta(a,k) = sup_{m \in \mathbb{Z}}\{a - (a + m)k\}$ for each real number $k$, where $\mathbb{Z}$ is the set of all integers. If $\alpha_{n} \geq 1 + \delta(a,\alpha_{n})/t $, for each nonnegative integer $n$, where $0 \leq a < 1$, $ 0 < t < 1$, and $b = a + t$, then it is proved that there exists a $\xi \in [a+m,b+m]$, for each $m \in \mathbb{Z}$, such that $0 \leq \{\xi \beta_{n} - a\} \leq t$ holds for all nonnegative integers $n$. Further, if $t\alpha_n\alpha_{n+1} -(1+\delta(a,\alpha_n))\alpha_{n+1} -t-\delta(a,\alpha_{n+1}) \geq 0$ for infinitely many nonnegative integers $n$, then for each $m \in \mathbb{Z}$, there exists a set of $\xi \in [a+m,b+m]$ that has the cardinality of the continuum so that $0 \leq \{\xi \beta_{n} - a\} \leq t$ is true for all nonnegative integers $n$.

In 1987 Jarník and Kurzweil [11] proved the following result: \emph{A function $F:[a,b] \to {\mathbb R}$ is $AC^*G$ on $[a,b]$ if and only if $\mu_F^*$ (Thomson's variational measure) is absolutely continuous on $[a,b]$ and $F$ is derivable $a.e.$ on $[a,b]$.} But condition $``F$ is derivable $a.e.$ on $[a,b]$'' is superfluous, as it was shown in \cite{Ene19}. In this paper we shall improve this result (from where we obtain an answer to a question of Faure [9]. Then using Faure's definition for a Kurzweil-Henstock-Stieltjes integral with respect to a function $\omega$, we give corresponding definitions for: a Denjoy$^*$-Stieltjes integral with respect to $\omega$, a Ward-Perron-Stieltjes integral with respect to $\omega$, a Henstock-Stieltjes variational integral with respect to $\omega$, and we show that the four integrals are equivalent.

In this paper, using Thomson's local systems, we introduce some very general integrals, each containing the wide Denjoy integral: the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal D}]$-integral (of Lusin type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal V}]$-integral (of variational type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}]$-integral (of Ward type); the $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal R}]$-integral (of \linebreak Riemann type); We prove that in certain conditions the integrals $[{\mathcal S}_1 \!{\mathcal S}_2 {\mathcal V}]$ and $[{\mathcal S}_1 {\mathcal S}_2 {\mathcal W}]$ are equivalent (it is shown that the first integral satisfies a Saks-Henstock type lemma). For the $[{\mathcal S}_1{\mathcal S}_2{\mathcal R}]$-integral we only show that it satisfies a quasi Saks Henstock type lemma (see Lemma 7.4). Finally, if ${\mathcal S}_1 = {\mathcal S}_o^+$ and ${\mathcal S}_2 = {\mathcal S}_o^-$ we obtain that the integrals $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal V}]$, $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal W}]$ and $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}]$ are equivalent (in fact the $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal D}]$-integral is exactly the wide Denjoy integral). But the equivalence of the three integrals with the $[{\mathcal S}_o^+ {\mathcal S}_o^- {\mathcal R}]$-integral follows only if we assume the additional condition that the primitives of the $[{\mathcal S}_o^+{\mathcal S}_o^- {\mathcal R}]$-integral are continuous (see Theorem11.1).

We consider two definitions of upper porosity of measures and we prove that the first one only can take the values o and $\frac{1}{2}$ and the second one, the values of 0, $\frac{1}{2}$. or 1.

The classical Black-Scholes-Merton method for pricing European call options uses the Itô calculus to model the processes involved. We show how to model stochastic process using Henstock integrands instead of Itô differentials (or stochastic integrals), and we show how to derive the Black-Scholes partial differential equation and pricing formulae using elementary methods.

We compute the Hölder spectrum and the moment sums of the typical (in the sense of Baire category) monotone continuous function, $f$. We show that these functions are of multifractal nature, for example $\alpha$ equals the Hausdorff dimension of those points where such a function, $f$ is of exact Hölder class $\alpha\in [0,1].$ We also prove that $f$ "grows" only at those points where it has bad Hölder properties, that is, at those points where it is of Hölder class $0.$ The upper moment sum $\overline{\tau}_f (q)=0$ if $q <0,$ $\overline{\tau}_f (q)=0=1-q$ if $0\leq q\leq 1,$ and $\overline{\tau}_f (q)=0$ if $1< q.$ The lower moment sum $\underline{\tau}_f (q)=1-q$ if $q<0,$ $\underline{\tau}_f (q)=0$ if $0\leq q\leq 1,$ and $\underline{\tau}_f (q)=1-q$ if $1<q.$

We introduce an abstract treatment of the weak convergence for bounded monotone set functions which allows us to obtain some basic results generalizing well known theorems regarding classical weak and vague convergence and weak convergence of masses on normal topological spaces (e.g. Portmanteau type theorems, Direct and Converse Prokhorov type theorems). Moreover, we introduce a suitable topology (called the L\'evy-topology) in order to study the properties of this abstract convergence from a topological point of view.\newline

We consider two types of topological spaces which generalize the notion of a Lindelöf space. The invariance of these classes under some operation is discussed. We also show that the density topology in $\mathbb{R}^n$ is an example of both generalizations.

It is known that the six Darboux-like function spaces of continuous, extendable, almost continuous, connectivity, Darboux, and peripherally continuous functions $f\:R\to \R$, with the metric of uniform convergence, form a strictly increasing chain of subspaces. We denote these spaces by \C, \E, \AC, \Conn, \D, and \PC, respectively. We show that C and D are porous and AC and Conn are not porous in their successive spaces of this chain.

For continuous self maps of $[0,1]$, we extend M. K. Fort, Jr.'s notion of an essential fixed point to points generating nonsingleton $\omega $-limit sets. The $\omega $-limit sets of these essential points are, in a metric sense, stable under small perturbations of the function. We develop some of the properties of the essential point set of a continuous function, and investigate the relationship between essential points, $\omega $-limit sets, and the chaotic nature of the generating function.

Necessary and sufficient conditions are found for the positive Borel measure $\nu$, which provide the boundedness (compactness) of the generalized Riemann--Liouville operator from one Lebesgue space into another Lebesgue space with measure $\nu$. The appropriate problem for the generalized Weyl operator is solved as well.

By means of a certain well known family $\mathcal {B}$ of subsets of $\mathbb{R}$ fulfilling two conditions we introduce some topologies on $\mathbb{R}$ (in Section 2 we consider the density topology). We observe that the family of the sets $\Omega _f (y):= \left\{ x \in \mathbb{R};\; \; \omega _f(x) \geq y \right\}$ for an arbitrary bounded function $f\:\mathbb{R} \to \mathbb{R}$ (where $\omega _f(x)$ is a kind of ${\cal B}$-oscillation of $f$) has three properties. Then we show that for each family $\left\{ \Omega (y) \right\} _{y \in [0,1]} \subset 2 ^{\mathbb{R}}$ having similar properties and in addition fulfilling conditions $M _1$ and $\mathcal {U}^\prime $ (known from the literature) there is a function $f: \mathbb{R} \to [0,1]$ such that $\Omega _f(y) = \Omega (y)$ for each $y \in[0,1]$. In Section 2 we prove some analogous result for the density topology.

The present paper deals with a certain property of multivalued functions which coincides with the Darboux property in the case of single valued real functions of real variable. It is well known that derivatives, and hence approximately continuous functions have the Darboux property. The results contained here are generalizations of these properties to the multivalued case.

We extend additive set-valued set functions and normal multimeasures defined on a ring of subsets. We also prove a Carathéodory-Hahn-Kluvanek-type theorem for additive set-valued set functions. Finally, we establish results on the extension of transition multimeasures.

We obtain information about the structure of nonconvex compacta $C$ in $\R^n$ ($n\ge3$) having the property that every projection onto a hyperplane is convex. We find that if $\dim(C)\le n-2$, then $C$ contains $n+1$ copies of $S^{n-2}$ that are contained in distinct hyperplanes. If $\dim(C)\ge n-1$, then the existence of three and no more than three hyperplanes that intersect $C$ in an $(n-2)$-sphere can be guaranteed.

We study $C (X) \cap L^p (X)$, the set of all continuous functions in $L^p (X)$, as a subspace of $L^p (X)$, and show that it is $\Pi^0_3$-complete when $X$ is Polish locally compact, while it is $\pi^1_1$-complete when $X$ is Polish not $\sigma$-compact. We also show that the subspace of Riemann integrable functions and, for every $k =1,2, \dots, \infty$, $C^k (\mathbb{R})$ are $\Pi^0_3$-complete in $L^p (\mathbb{R})$. In contrast the subspace of all everywhere differentiable functions is $\pi^1_1$-complete in $L^p (\mathbb{R})$. If $X$ is locally compact, we consider $C (X)$ endowed with the compact-open topology and establish the complexity of some of its subspaces, including $C_0(X)$, $C_{00}(X)$ and $UC (X,d)$.

We establish a special representation of uniformly bounded martingale differences and give some examples of its application. In particular, we prove that any uniformly bounded ONS $\left\{ f_{k}\right\}$ contains a subsystem $\left\{ f_{k_{n}}\right\} $ ($k_{n}\leq 2^{n}$) of\ \ unconditional convergence with exponential estimation of majorants.

This paper presents one interesting metric discovered while studying convergence of series with alternating signs. In previous papers [D1] and [D2] we presented a study of 'typical' series with alternating signs. Namely, given a sequence of real nonnegative numbers whose sum is infinity we consider all different ways the signs plus or minus could be put in front of each of these numbers. With a given metric we ask what is the 'size' of the set of those choices of $+$ or $-$ for which the resulting series with alternating signs converges. The term 'size' here means either the Baire category or porosity of this set. While metrics studied before as Frèchet, Baire or Euclidean allowed us to get interesting results, they all share one undesirable property - insensitiveness to the change of a 'tail' of a sequence. The D-metric, introduced here, does not have this undesirable property.

It is known that the concepts of continuous norm and of absolutely continuous norm do not coincide. There exists a space in which all functions possess continuous norm but not all functions possess absolutely continuous norm. In this paper we construct an extremal example of a Banach function space in which all functions have continuous norm but only the zero function has absolutely continuous norm.

We prove theorems concerning common divisor for the families of the quotients of Darboux functions with respect to Darboux property.

We introduce an $n$-dimensional Perron integral defined in terms of ordinary (in the Saks terminology) derivates. We prove that we get an equivalent definition of this integral if in the definition we use only continuous major and minor functions. We also prove that this integral is equivalent to a version of the Kurzweil-Henstock integral defined by Mawhin.

This note contains a proof of the Fundamental Theorem of Calculus for the Lebesque-Bochner integral using Hausdorff measures (see 2.4). For the real case $(X=\mathbb{R})$, this proof uses only the basics from the Lebesque integral theory (see 2.6).

Let $g:X \rightarrow X$ be a bijection and $n\geq 2$ be a fixed integer. We consider the equation of iterative roots $\varphi ^n(x)=g(x)$ and we look for its solution with big graph: big from the point of view both of topology and measure theory.

In the thirties, Banach, Mazurkiewicz and Jarn\'{\i}k found relations connecting Dini derivates of a typical continuous function on $[0,1]$ at all points of $(0,1)$. We prove, answering a question of K. M. Garg, that there are no further relations of this sort. An analogous result is proved also for approximate Dini derivates. The aim of this note is to present relatively simple proofs of these results. An article containing an improvement of these results in several directions (with a considerably more complicated proof) is in preparation.

Consider a $\sigma$-ideal, $\sigma$-algebra pair $\cal{I} \subseteq \cal{A}$ on a Polish space $X$ which has no isolated points, such that $\cal{A}$ contains all the Borel subsets of $X$ while $\cal{I}$ contains all the countable subsets of $X$, but none of the perfect subsets of $X$. We show that if $(\cal{I}, \cal{A})$ admits a simultaneous MB-like characterization consisting of Borel sets, then $(\cal{I},\cal{A})$ is $((s_{0}), (s))$, the $\sigma$-ideal, $\sigma$-algebra pair of Marczewski null, Marczewski measurable sets. We deduce some results about uniformly completely Ramsey sets.

Some properties of finite continuous functions are investigated. In particular we show that, in the class of Darboux functions, the family of 2-continuous functions is the same as the family $B_1^{**}$ and the set of all discontinuity points of finite continuous, Darboux functions is nowhere dense.

This paper gives three pair of adjoint classes of the Lebesgue-Stieltjes integrable functions.

In this article we show that for the discrete limit $f$ of sequence of bilaterally quasicontinuous Baire 1 functions the complement of the set of all points at which $f$ is bilaterally quasicontinuous and has Darboux property, is nowhere dense. Moreover, a construction is given of a bilaterally quasicontinuous function which is the discrete limit of a sequence of Baire 1 functions, but is not the discrete limit of any sequence of bilaterally quasicontinuous Baire 1 functions.

We consider a special condition implying the strong product measurability of bounded functions with values in a Banach space which all vertical sections are derivatives.

We present an elementary proof that the graph of a continuous function from $\mathbb{R}$ to $\mathbb{R}$ is not purely unrectifiable. As a consequence of our method, we observe that all continuous functions from $\mathbb{R}$ to $\mathbb{R}$ meet the graph of some monotonic function in a set of positive linear measure.

Georganopoulos (see [1] ) has shown that a continuous function $% f:X\rightarrow B$, where $X$ is a compact metric space and $B$ a convex subset of a real normed space $Y$, is the uniform limit of a sequence of Lipschitz maps from $X$ to $B$. In this note we obtain a similar result, namely we show that a continuous function $f:X\rightarrow R$, where $X$ is a metric space, is a uniform limit of a sequence of locally Lipschitz maps from $X$ to $R$. When $X$ is compact and $B=Y=R$, we get the Georganopoulos's result.

Let $G$ be a topological group of second category, ${\cal B}_G$ be its Borel $\sigma$-algebra and ${\cal B}$ a $\sigma$-algebra of subsets of $G$ such that $(G,{\cal B})$ is a measurable group. For a probability measure $P$ on $(G,{\cal B}),$ let $P_g(E):=P(gE)$ for $g\in G, E\in{\cal B}.$ The aim of this note is to show that there exists an inner-regular probability measure $P$ and a $\sigma$-finite measure $\mu$ on $(G,{\cal B})$ such that $P_g\ll \mu\; \forall\; g\in G,$ iff $G$ is locally-compact and in that case $P_g\ll \lambda_G\ll \mu\;\;\forall\; g\in G$ on the $\sigma$-algebra ${\cal B} \cap {\cal B}_G,$ where $\lambda_G$ denotes a Haar measure of $G.$

The principal measure of chaos of a continuous map of an interval is generated by a pair of points at least one of which belongs to an omega limit set of the second kind if such set exists, or is zero in the case that such set does not exist.

If $\sum a_{k}$ and $\sum b_{k}$ are series of non-negative terms, we provide a necessary and sufficient condition that $sup_{n}\{% \sum_{1}^{n}a_{k}/\sum_{1}^{n}b_{k}\}=\infty .$

This note concerns the generalized Riemann integrals using free tagged subdivisions, and leading to absolute integration theories. We first establish an approximation theorem by step functions. This result will allow us to define a natural notion of equi-integrability in Lebesgue space $L^1$. Some applications, as a new caracterization of compact parts of $L^1$, are presented.

We note that the restriction of any measurable mapping $f\:R\to\R^n$ to the set of points at which it possesses a finite approximate derived number maps Lebesgue null sets to sets of zero linear measure. As a corollary we deduce an optimal version of Denjoy-Young-Saks's theorem for approximate derivatives valid up to exceptional sets of zero linear measure in the graph.