Recently a connection has been found between the improper Kurzweil-Henstock integral on the real line and the integral over a compact space. In this paper these results are extended to a Pettis-type integral for the case of functions with values in Riesz spaces with ``enough" order continuous functionals.

We prove general results from which large families of pairwise disjoint, Borel inseparable, complete coanalytic sets can be obtained. The elements of such families are naturally indexed by $2^{\aleph _0}$, $\omega _1$ or the classes of a Borel complete equivalence relation. We also give some concrete examples of such families in analysis, topology etc.

Let $f$ be an interval map defined on a neighborhood of a fixed point $0$ with \(f'(0)=\lambda\) where \(0<|\lambda|<1\) and let \(\phi_{n}(x)=f^{n}(x)/\lambda^{n}\). It is shown that if \[f(x)=\lambda x+\mathcal{O}\!\left(\frac{|x|}{y\log{(y)}\cdots \log^{p-1}{\!(y)}(\log^{p}(y))^{1+\varepsilon}}\right)\] for some \(\varepsilon>0\) and nonnegative integer $p$ where \(y=|\!\log{(|x|)}|\), then the Koenigs' sequence \(\{\phi_{n}\}\) of $f$ converges uniformly on a neighborhood of $0$ to a limit \(\phi\) with \(\phi(0)=0\) and \(\phi'(0)=1\). On the other hand, if $f(0)=0$ and \[f(x)=x\!\left(\!\lambda -\frac{1}{\log(x)\log(-\log(x)) \cdots\log^{p}{(-\log(x))}}\right)\] for sufficiently small $x>0$ where $0<\lambda<1$ and $p$ is a nonnegative integer, then the Koenigs' sequence of $f$ diverges on a small right-neighborhood of $0$. It is illustrated by examples that when \(\varepsilon=0\) in the first equation for $f$ given above, the Koenigs' sequence of $f$ can also converge to zero on a neighborhood of $0$ or converge to a limit \(\phi\) that is nondifferentiable at $0$. It is also shown that when the Koenigs' sequence of a map $f$ converges to a limit \(\phi\) that is differentiable at $0$, then \(\phi'(0)\) is either $0$ or $1$.

In this note a particular case of the following general problem is considered: how to control lower order derivatives by higher ones, at least over a sequence of points. The following particular case is proved: if a $C^2$ negative-valued function $h=h(w)$ depends on one complex variable in the unit disc and $h(1)=h_w(1)=0$, then the first derivative $h_w$ is controlled by the Laplacian of $h$ over a sequence of points converging to $w=1$. Such kind of estimates have applications to delicate problems of convexity with respect to various families of functions.

In this article we establish some conditions concerning the sections $f^y$ of a function $f:{\mathR}^2 \to {\mathR}$ having Lebesgue measurable sections $f_x$ which imply the measurability of $f$. The first condition is more general than condition ${\cal A}$ introduced in \cite{5}.

A function $f:{\mathR} \to {\mathR}$ satisfies the condition $(s_1)$ if for each real $r > 0$, for each $x$, and for each set $U \ni x$ belonging to the density topology there is an open interval $I$ such that $C(f) \supset I \cap U \neq \emptyset $ and $f(U\cap I) \subset (f(x)-r,f(x)+r)$. ($C(f)$ denotes the set of all continuity points of $f$). In this article we investigate the sums of two Darboux functions satisfying the condition $(s_1)$.

We investigate the connection between the Borel class of a function $f$ and the Borel complexity of the set $\T(f)=\{C\in\comp(X)\colon f\rest_C\text{ is continuous}\}$ where $\comp(X)$ denotes the compact subsets of $X$ with the Hausdorff metric. For example, we show that for a function $f\colon X\to Y$ between Polish spaces; if $\T(f)$ is $F_{\sigma\delta}$ in $\comp(X)$, then $f$ is Borel class one.

In this paper, we prove that if $f$ is Henstock-Kurzweil integrable on a compact subinterval $[a,b]$ of the real line, then the following conditions are satisfied: (i) there exists an increasing sequence $\{X_n\}$ of closed sets whose union is $[a,b]$; (ii) $\{f{\chi_{ _{X_n}}}\}$ is a sequence of Lebesgue integrable functions on $[a,b]$; (iii) the sequence $\{f{\chi_{ _{X_n}}}\}$ is Henstock-Kurzweil equi-integrable on $[a,b]$. Subsequently, we deduce that the gauge function in the definition of the Henstock-Kurzweil integral can be chosen to be measurable, and an indefinite Henstock-Kurzweil integral generates a sequence of uniformly absolutely continuous finite variational measures.

We study some conditions to guarantee the existence of common fixed points of two commuting Cournot maps $F(x,y)=(f_{2}(y),f_{1}(x)),$ $G(x,y)=(g_{2}(y),g_{1}(x)),$ defined from $I^{2}=[0,1]^{2}$ into itself. In particular, we prove that Jungck's Theorem and Jachymski's equivalent conditions can be only partially proved in this setting.

We characterize both the family of the sums of nonnegative Darboux/almost continuous functions (in the sense of Stallings) and the family of the products of bounded Darboux/almost continuous functions.

The main goal of this paper is to characterize the family of functions which are the averages of two comparable almost continuous functions.

We characterize the family of all $\Hj$-integrable functions and solve several problems related to the $\Hj$-integral.

We show that, for each pair of metric spaces that has the Lipschitz extension property, every bounded uniformly continuous function can be approximated by Lipschitz functions. The same statement is valid for functions between a locally convex space and $\mathbb{R}^{n}$. In addition, we show that for a locally bounded, convex function $F:X\rightarrow\mathbb{R}^{n}$, where $X$ is a separable Fréchet space, the set of points on which the differential of this function exists is dense in $X$.

We show that the class $SZ$ of Sierpinski-Zygmund functions has a nonempty intersection with the class $\extb$ of all uniform limits of sequences of extendable connectivity functions $f_n:\R\to\R.$ We reconsider the idea of $f$-negligible sets this time with respect to $f\in \extb.$ We also show that under MA, $SZ\cap \extb$ cannot be characterized by preimages of sets.

The purpose of this paper is to give a simple and completely elementary proof that the derivative of the indefinite Riemann-complete integral equals the integrand almost everywhere. Elementary in this case means using the least amount of measure theory possible.

For a harmonic function on an open subset of real $n$-space, we propose a condition on the Taylor expansion that implies harmonic extension to a larger set, by a result on the size of the domain of convergence of its Taylor series. The result in the $n=2$ case is due to M.\ B\^ocher (1909), and the generalization to $n>2$ is given a mostly elementary proof, using basic facts about multivariable power series.

We consider the question of measurability of functions obtained by using the superposition operator which is induced by a given function of two variables. Some related measurability properties of functions of two variables are also discussed.

We give certain characterizations of the so-called natural functions introduced by Chistyakov. We show that the set of natural functions from $[a, b]$ into a normed space $X$ is small in the porosity sense in the set of uniform Lipschitz 1 functions with the norm inherited from the space of $BV$ functions. The sizes of other classes of Lipschitz functions in $BV$ are also established.

We say that a set $X \sq \real^2$ is {\it Sierpi{\'n}ski-Zygmund\/} (or {\it SZ-set\/} for short) if it does not contain a partial continuous function of cardinality continuum $\cont$. We observe that the family of all such sets is $\cf(\cont)$-additive ideal. Some examples of such sets are given. We also consider {\it SZ-shiftable sets\/}; that is, sets $X \sq \real^2$ for which there exists a function $f\colon \real \to \real$ such that $f+X$ is a SZ-set. Some results are proved about SZ-shiftable sets. In particular, we show that the union of two SZ-shiftable sets does not have to be SZ-shiftable.

We note a gap in Sciffer's construction of an everywhere irregular Lipschitz function on the line and provide a different simple construction of such a function, which even reaches maximal irregularity at every point.

Let $E \subseteq {\mathbb R}$ and $g:E \to {\mathbb R}$. We show that if $|g(E)| = 0$, then $\underline{g}_{\,ap}(x) \le 0 \le \overline{g}_{ap}(x)$ almost everywhere on $E$, which immediately implies a lemma of Krzyzewski \cite{10} and Foran \cite{7}. The function $g$ is said to satisfy the inverse Lusin condition $(N^{-1})$ on $E$ if $|g^{-1}(H)| = 0$ for every $H \subseteq g(E)$ with $|H| = 0$. We prove that if $g^\prime_{ap}(x)$ exists almost everywhere on $E$, then $g$ is an $N^{-1}$-function if and only if $g_{ap}^\prime(x) \neq 0$ almost everywhere on $E$. We also improve upon Foran's \cite{7} chain rule for approximate derivatives, and obtain necessary and sufficient conditions for its validity almost everywhere.

We present a construction of an uncountable subset of the reals which belongs to every \s-ideal $I$ on ${\mathbb R}$\ with the property that there is no uncountable family of disjoint Borel sets outside $I$.