We continue the investigation began in [1] of the connection between the structure of a function $f$ defined on a subset of a space $X$ and the Borel complexity of the set $C(f) =$ {$C \in J(X): f| C$ is continous where $J(X)$ denotes the nonempty compact subsets of $X$ with the Hausdorff metric. Two hierarchies of functions with $G_{\delta}$-graphs are defined. We conjecture that they coincide

We show that functions of two variables whose vertical sections are preponderantly continuous are strongly supmeasurable.

It is shown that a function $f : \mathbb{R} to \mathbb{R}$ is Darboux and upper semicontinuous if and only if its maximum with each almost continuous function is almost continuous. This result generalizes an old theorem due to J. Farková.

Let $C([0,1])$ be the set of all continuous functions mapping the unit interval $[0,1]$ into $\mathbb{R}$. A function $f\in C([0,1])$ is called Besicovitch if it has nowhere one-sided derivative (finite or infinite). We construct a set $\mathcal{B}_{\sup}\negthickspace\subset C([0,1])$ such that $(\mathcal{B}_{\sup},\vert\vert~\vert\vert_{\sup})$ is an infinite dimensional Banach (sub)space in C([0,1]) and each nonzero element of $\mathcal{B}_{\sup}$ is a Besicovitch function.

We prove that the problem of the existence of a discontinuous separately continuous function $f:X \times Y\to\mathbb{R}$ for any non-discrete Tychonov spaces $X,\,Y$ of countable pseudocharacter is equivalent to NCPF (Near Coherence of $P$-filters) which is independent of ZFC. Also for every non-discrete Tychonov space $X$ we find an abelian topological group $G$ of countable cellularity and a discontinuous separately continuous function $f:X \times G \to\mathbb{R}$.

Wilczyński's definition of the Lebesgue density point given in [W1] opened the possibility of more subtle properties of the notion of density point and the density topology, their various modifications and most of all category analogues. In this paper we introduce a notion of an $\mathcal{A} _{d}-$density point of a measurable set on the real line. The notion is a generalization of Lebesgue density and is based on the definition given by Wilczyński. We prove that the $\mathcal{A}_{d}-$density topology generated by this notion is strictly finer than the Lebesgue density topology and we examine several of its properties.

In this paper we define the notion of order of a function, which measures its growth rate with respect to a given function. We introduce the notions of continuity and linearity at infinity with which we characterize order-comparability and equivalence. Using the theory we have developed, we apply orders of functions to give a simple and natural criterion for the uniqueness of fractional and continuous iterates of a function.

It is proved that every derivation is the sum of two almost continuous (in Stallings' sense) derivations and the limit of a sequence (of a transfinite sequence) of almost continuous derivations.

Let $I=[0,1]$, and let $\mathcal J$ be the class of functions $I \rightarrow I$ with connected $G_{\delta}$ graph. Recently it was shown that dynamical systems generated by maps in $\mathcal J$ have some nice properties. Thus, the Sharkovsky's theorem is true, and a map has zero topological entropy if and only if every periodic point has period $2^{n}$, for an integer $n\ge 0$. In this paper we consider, for a map $\varphi$ in $\mathcal J$, properties of $\omega$-minimal sets; i.e., sets $M\subset I$ such that the $\omega$-limit set $\omega_{\varphi}(x)$ is $M$, for every $x \in M$. If $\varphi $ is continuous, then, as is well-known, $M$ is minimal if and only if $M$ is non-empty, closed, $\varphi(M)\subseteq M$, any point in $M$ is uniformly recurrent, and no proper subset of $M$ has this property. In this paper we prove that the same is true for $\varphi\in\mathcal J$ with zero topological entropy, but not for an arbitrary $\varphi\in\mathcal J$.

An integral, called the $Bk$-integral, for functions taking values in a locally convex space is defined. Properties of $Bk$-integrable functions are considered and the relations with other integrals are studied. Moreover the $Bk$-integrability of bounded functions is compared with the Bourgain property.

Let $\nu$ be a finite Borel measure on $[0,1]^d$. Consider the $L^q$-spectrum of $\nu$: $\tau_\nu(q)=\liminf_{n\to\infty}-n^{-1} \log_b\sum_{Q\in\mathcal{G}_n}\nu (Q)^q$ ($q\ge 0$), where $\mathcal{G}_n$ is the set of $b$-adic cubes of generation $n$ ($b$ integer $\ge 2$). Let $q_\tau=\inf\{q\: \tau_\nu(q)=0\}$ and $H_\tau=\tau_\nu'(q_\tau^-)$. When $\nu$ is a mono-dimensional continuous measure of information dimension $D$, $(q_\tau,H_\tau)=(1,D)$. When $\nu$ is purely discontinuous, its information dimension is $D=0$, but the pair $(q_\tau,H_\tau)$ may be non-trivial and contains relevant information on the distribution of $\nu$. Intrinsic characterizations of $(q_\tau,H_\tau)$ are found, as well as sharp estimates for the large deviation spectrum of $\nu$ on $[0,H_\tau]$. We exhibit the differences between the cases $q_\tau=1$ and $q_\tau\in (0,1)$. We conclude that the large deviation spectrum's properties observed for specific classes of measures are true in general.

Let $ m $ be a finite Borel measure on $[0,1]^d$. Consider the $L^q$-spectrum of $ m $: $\tau_ m (q)=\liminf_{n\to\infty}-n^{-1}\log_b\sum_{Q\in\mathcal{G}_n,\ m (Q)\neq 0} m (Q)^q$, where $\mathcal{G}_n$ is the set of $b$-adic cubes of generation $n$. Let $q_\tau=\inf\{q\: \tau_ m (q)=0\}$ and $H_\tau=\tau_ m '(q_\tau^-)$. When $ m $ is a mono-dimensional continuous measure of information dimension $D$, $(q_\tau,H_\tau)=(1,D)$. When $ m $ is purely discontinuous, its information dimension is $D=0$, but the non-trivial pair $(q_\tau,H_\tau)$ may contain relevant information on the distribution of $ m $. The connection between $(q_\tau,H_\tau)$ and the large deviation spectrum of $ m $ is studied in a companion paper. This paper shows that when a discontinuous measure $ m $ possesses self-similarity properties, the pair $(q_\tau,H_\tau)$ may store the main multifractal properties of $ m $, in particular the Hausdorff spectrum. This is observed thanks to a threshold performed on~$ m $.

We consider convergence (introduced by E. Wagner in 1981) with respect to a $\sigma$-ideal of $\mathscr{S}$-measurable real valued functions on $Y$ where $\mathscr{S}\subset \mathscr{P} (Y)$ is a $\sigma$-algebra containing a given $\sigma$-ideal $\mathscr{J}$. We check which operations preserve completeness generated by convergence with respect to a $\sigma$-ideal. We introduce uniform kinds of $\mathscr{J}$-convergence and $\mathscr{J}$-completeness and use them in a statement concerning the Fubini product of two $\sigma$-ideals.

et $A$ be a subset of a Banach space $X$ and $f$ a Fréchet differentiable function on $A$ (with respect to $A$). We give a simple proof of the connectedness of the graph of $f'$ in $X\times X^*$ under relatively weak conditions on $A$. In particular, we simplify a proof by J. Malý of the connectedness of the range of $f'$ for some convex sets $A$. At the same time, we extend an older result of C. E. Weil on the connectedness of the range of $f'$ for some non-convex sets $A\subset\mathbb R^n$.

We study integral inequalities for the Hardy operator $Hf$ of the form $\int_0^\infty\Phi[Hf^p]\,d\mu\leq c_0\int_0^\infty\Phi[c_1f^p]\,d\mu$, where $\Phi$ is convex, $\mu$ is a measure on $\mathbb R_+$, $1\leq p < \infty$, and $f$ is non-increasing. The results we obtain are extensions of the classical $B_p-$ weight theory [1,5].

In this article we investigate the maximal additive and maximal multiplicative families for the classes of quasicontinuous functions in the sense of Piotrowski and Vallin introduced in [8].

This paper introduces a few characterizations of quasi continuity of functions and minimality of multifunctions. The methods used are based on the concept of cluster systems and derived operations acting on multifunctions. Inclusion relations between original multifunction and resultant cluster multifunction describe properties connected with generalized continuities and minimality.

In this article we investigate the maximal additive and maximal multiplicative families for the classes of quasicontinuous functions in the sense of Piotrowski and Vallin introduced in [8].

In this paper, an attempt has been made to characterize one-sided points of cliquishness of a function in terms of generalized one-sided oscillations and study the nature of one-sided points of cliquishness of the quasi-uniform limit function of a sequence of functions.

This note concerns some variations on the classical Theorem of Banach-Zarecki[2].

It is shown that some analog of Minkowski's method in geometric number theory enables to establish the existence of sets which are absolutely nonmeasurable with respect to weakly metrically transitive invariant measures.

We show that every set of reals is a set of points of weak symmetry for some function $f : \mathbb{R} \to \mathbb{N}.

We present a general theory of absolutely continuous paths with values in metric spaces using the notion of metric derivatives. Among other results, we prove analogues of the Banach-Zarecki and Vallée Poussin theorems.

We discuss the Arzela-Ascoli pre-compactness Theorem from the point of view of Functional Analysis, using compactness in $\ell^\infty$ and its dual.

A function that maps intervals into intervals is called a Darboux function. We prove that if $g$ is a continuous function that is non-constant on every non-empty open interval, and $f$ is a Darboux function such that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}$, and the set of all such $n_{x}$ is bounded, then $f$ is continuous. In the above statement, the hypothesis ``the set of all such $ n_{x} $ is bounded'' cannot be dropped. We also show that if $g$ is a continuous function that takes a constant value $k$ on some non-empty open interval $I$ and $k\in I$, then there exists a discontinuous Darboux function $f\:mathbb{R}\rightarrow \mathbb{R}$ with the property that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}\leq 2$. In the previous statement, if $k\notin I$, then no conclusion can be drawn about the function $f$.

We demonstrate how certain types of symmetric derivatives originate from a simple least-squares regression problem involving discrete Chebyshev polynomials. As the number of data points used in this regression tends to infinity, the resulting integrals, which involve Legendre polynomials, lead to Lanczos derivatives, a result that demonstrates how this latter entity is merely a continuous version of the symmetric derivative.