We establish the existence of an entire weak solution for a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow-up at infinity. The proof relies on Chang's version of the Mountain Pass Lemma for locally Lipschitz functionals. Our result generalizes in a nonsmooth framework, a result of Rabinowitz [12] related to entire solutions of the Schrödinger equation.

Properties of the families of finitely continuous and strong finitely continuous functions are investigated. We show that the Darboux property implies continuity of strong finitely continuous functions and that the family ${ DB}_1^{**} $ is superporous in the space of all finitely continuous functions with the Darboux property.

We prove that, for any nonconstant continuous function $f,$ there exists a continuous N-function $g$ such that $f+g$ is not an N-function. This answers a query by F. S. Cater.

The paper is a follow-up to our previous work [P.~Sworowski, An answer to some questions of Ene, Real Analysis Exchange, 30(1) (2004/05), 183--192], where we have given a chart of relations between four approximately continuous Denjoy-Khintchine type integrals. Here we complete that chart with the approximate Foran integral.

We investigate the intersection conditions for an $\s$-density system of paths. We show, for example, that for every unbounded and nondecreasing sequence of positive numbers $\s$ such that $ \liminf_{n\rightarrow \infty }\frac{s_n}{s_{n+1}} =0$, there exists a system of paths connected with $\s$-density points which does not satisfy the intersection conditions. Moreover, we show that a function $f\:mathbb{R}\rightarrow \mathbb{R}$ is $\s$-approximately continuous if and only if $f$ is continuous with respect to some $\s$-density system of paths.

Let $f$ be a distribution (generalized function) on the real line. If there is a continuous function $F$ with real limits at infinity such that $F'=f$ (distributional derivative), then the distributional integral of $f$ is defined as $\int_{-\infty}^\infty f = F(\infty) - F(-\infty)$. It is shown that this simple definition gives an integral that includes the Lebesgue and Henstock--Kurzweil integrals. The Alexiewicz norm leads to a Banach space of integrable distributions that is isometrically isomorphic to the space of continuous functions on the extended real line with uniform norm. The dual space is identified with the functions of bounded variation. Basic properties of integrals are established using elementary properties of distributions: integration by parts, H\"older inequality, change of variables, convergence theorems, Banach lattice structure, Hake theorem, Taylor theorem, second mean value theorem. Applications are made to the half plane Poisson integral and Laplace transform. The paper includes a short history of Denjoy's descriptive integral definitions. Distributional integrals in Euclidean spaces are discussed and a more general distributional integral that also integrates Radon measures is proposed.

In this article we prove that some results of Richter and Stephani concerning the cluster sets of quasicontinuous and cliquish real functions ([6]) are also true for the special quasicontinuities introduced by Piotrowski and Vallin in ([5])

We find a decomposition of the type of Sobczyk-Hammer for measures with values in $l$-groups, and also deduce some convergence theorems for such decompositions. Our procedure is based on some theorems of the type of Vitali-Hahn-Saks, and on the so-called Stone extension method.

In this article we characterize the discrete limits of sequences of piecewise linear functions. A function $f:[0,1]\to \mathbb{R}$ is the discrete limit of a sequence of piecewise linear functions iff there are closed sets $A_n$ such that $[0,1] = \bigcup_nA_n$ and the restricted functions $f/A_n$ are linear.

In this paper we focus on the Lebesgue property of Banach spaces. A real Banach space $X$ is said to have the Lebesgue property if any Riemann integrable function from $[0,1]$ into $X$ is continuous almost everywhere on $[0,1]$. We obtain a partial characterization of the Lebesgue property, showing that it has connections with the asymptotic geometry of the space involved.

Weight characterization is obtained for the $L^p$-$X^q$ boundedness of the two-dimensional Hardy operator $(H_2f)(x_1,x_2)=\int_0^{x_1}\int_0^{x_2}f(t_1,t_2)\,dt_1\,dt_2$. By using a limiting procedure as well as by a direct method, the corresponding boundedness of the two-dimensional geometric mean operator $(G_2f)(x_1,x_2)=\exp\bigg(\dfrac{1}{x_1x_2}\int_0^{x_1}\int_0^{x_2}\ln f(t_1,t_2)\,dt_1\,dt_2\bigg)$ is obtained.

We show that, with respect to the dynamics of iteration, Darboux-like functions from $\mathbb{R}$ to $\mathbb{R}$ can exhibit some strange properties which are impossible for continuous functions. To be precise, we show that (i) there is an extendable function from $\mathbb{R}$ to $\mathbb{R}$ which is `universal for orbits' in the sense that it possesses every orbit of every function from $\mathbb{R}$ to $\mathbb{R}$ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function $f\:mathbb{R}\to \mathbb{R}$ such that for every $n\in \mathbb{N}$, $f^n$ is almost continuous and the graph of $f^n$ is dense in $\mathbb{R}^2$, in spite of the fact that all $f$-orbits are finite. To prove (i) we assume the Continuum Hypothesis.

Notions of density points and topologies associated with various set-theoretic ideals connected with Hausdorff measures are introduced and their properties investigated. Inclusions between the ideals and between the topologies are shown.

This paper gives examples of two quasicontinuous functions whose second iterates are discontinuous everywhere. It is well-known that every quasicontinuous function has a dense---indeed, residual---set of points of continuity; our counter-examples show that this property does not hold for compositions of such functions.

In this paper, we show that if an R-integrable (or BV integrable function) defined over a compact interval $[a,b]\times [c,d]\subset \mathbb R^2$ satisfies certain conditions, then over any subinterval of $[a,b]\times [c,d],$ the iterated integral exists and equals the double integral. We also present some examples relevant to our theory.

Wilczyński's definition of Lebesgue density point given in [19] created a new tool for the study of the more subtle properties of the notion of density point and the density topology, their various modifications and most of all category analogues. In the paper we develop further properties of the $\mathcal{A}_{d}$-density topology on the real line, introduced in [22]. The topology is a generalization of the Lebesgue density topology and is based on the definition given by Wilczyński. We consider the properties of continuos functions with respect to the $\mathcal{A}_{d}$-density topology and prove that the topology is completely regular but not normal.

Let $S \subset R^2$ be the attractor of the iterated function system $\{f_1,f_2,f_3, f_4\}$, where $f_i(x)=\lambda_i x+b_i$, $i=1,2,3,4$, $x=(x_1,x_2) \in R^2$, $0<\lambda_i \le \frac{1}{2+\sqrt{2}}$,$b_1=(0,0)$, $b_2=(1-\lambda_2, 0)$, $b_3=(1-\lambda_3,1-\lambda_3)$, and $b_4=(0, 1-\lambda_4)$. This paper determines the exact centered covering measure of $S$ under some conditions relating to the contraction parameters.

The Bernstein-Bezier polynomials are known to possess total variation and length diminishing properties in one variable. We investigate the two dimensional generalizations to the square and the triangle. Simple counterexamples show that they do not diminish surface area. We consider Kantorovitch polynomials which seem to be a better choice to be area diminishing. A counterexample is given for the square. We then define the Kantorovitch polynomials on the triangle and give an area estimate for them.

We investigate measurable sets of an outer measure defined using ``tubes,'' and prove that the only tube-measurable sets are the tube-null sets and their complements.

In this paper, we obtain some characteristic conditions which are equivalent with the validity of the ``reverse'' Hardy inequality $(-\infty<q\leq p<0)$ and compare these characterizations.

Under the Continuum Hypothesis, it is shown that there exists a nonseparable extension of the Lebesgue measure on the real line whose nullsets coincide with the nullsets in the Lebesgue sense.

Let $\Omega $ be a domain in $\mathbb{R}^{n}$ with finite Lebesgue measure and $f\in C\left( \Omega \right) \cap L^{1}\left( \Omega \right) $ a real-valued function on $\Omega $. It is shown that there exists a unique number $M\in \mathbb{R}$ at which the function $I\left( y\right) =\int_{\Omega }\left| f\left( x\right) -y\right| d\lambda ^{n}\left( x\right) $ is minimized, where $\lambda ^{n}$ is the Lebesgue measure on $% \mathbb{R}^{n}$. We can define this number as \textit{the }median of $f$ over $\Omega $ with respect to $\lambda ^{n}$.