We give an improvement of the result that if $hP\ge0$ on $\R^n$, where $h$ is a harmonic function and $P$ a non-trivial harmonic polynomial, then $h$ is proportional to $P$.

Our purpose in this article is to prove that the equivalence relation of being of the same kind is not classifiable by countable structures.

Let $U$ be a locally compact Hausdorff space that is not compact. Let $L(U)$ denote the family of continuous real valued functions on $U$ such that for each $f\in L(U)$ there is a nonzero number $p$ (depending on $f$) for which $f\!-\!p$ vanishes at infinity. Then $L(U)$ is obviously a lattice under the usual ordering of functions. \par In this paper we prove that $L(U)$, as a lattice alone, characterizes the locally compact space $U$. \par Let $S$ be a locally compact Hausdorff space. Define $T(S)$ to be $L(S)$ if $S$ is not compact, and $T(S)$ to be $C(S)$ if $S$ is compact. We prove that any locally compact Hausdorff spaces $S_1$ and $S_2$ are homeomorphic if and only if their associated lattices $T(S_1)$ and $T(S_2)$ are isomorphic.

A family of multidimensional generalized Perron type integrals is constructed. It is shown that these integrals solve the problem of recovering, by generalized Fourier formulae, the coefficients of multiple Haar and Walsh series of some class. This class includes in particular series convergent $\rho$-regularly everywhere except some countable set $E \subset G^{d}$. It is shown that some properties of rectangularly convergent multiple Haar and Walsh series do not hold for the $\rho$-regular convergence.

We prove that for any continuous real valued function $f$ on $[a,b]$ there exists a continuous function $K$ such that $K\!-\!f$ has bounded variation and $(K\!-\!f)^\prime = 0$ almost everywhere on $[a,b]$ and such that in any subinterval of $[a,b]$, $K$ has right derivative $\infty$ at continuum many points, $K$ has left derivative $\infty$ at continuum many points, $K$ has right derivative $-\infty$ at continuum many points, and $K$ has left derivative $-\infty$ at continuum many points. Furthermore, functions $K$ with these properties are dense in $C[a,b]$. We can assume the infinite derivatives of $K$ are bilateral if $f$ is of bounded variation on $[a,b]$ or if $f$ satisfies Lusin's condition $(N)$.

We show that there exist Polishable subspaces of arbitrarily high Borel class in every infinite-dimensional separable Banach space.

Quasicontinuity with respect to one coordinate and symmetrical quasicontinuity strengthen the concept of classical quasicontinuity of a bivariate function $f$ from a product space $X \times Y$ into a topological space~$Z$. For certain spaces $X,Y$, we show that a function $f$ from $X \times Y$ into a metric space $Z$ is quasicontinuous with respect to the first coordinate if and only if it is the uniform limit of step functions quasicontinuous with respect to the first coordinate. This applies in particular to arbitrary $X \subseteq {\mathbb R}^m$, $m \ge 0$, and every $Y \subseteq {\mathbb R}^n$, $n \ge 1$, without isolated points. A second result concerns spaces $X,Y$ such that every continuous $f:X \times Y \rightarrow Z$ is the uniform limit of symmetrically quasicontinuous step functions. It comprises all $X,Y \subseteq {\mathbb R}$ without isolated points.

The quadruplet $(C(f), Q(f), E(f), A(f))$ is characterized, where $C(f)$, $Q(f)$, $E(f)$ and $A(f)$ are the sets of all continuity, quasicontinuity, upper and lower quasicontinuity and cliquishness points of a real function $f$ of real variable, respectively.

We extend the notion of finite statistical variation of single sequences by Faisant et al (2005) to double sequences using the double natural density of the set $\mathbb{N} \times \mathbb{N}$. Certain consequences of this notion are investigated including its relation with statistical convergence introduced earlier by Mursaleen and Edely (2003). We also introduce the more general concept of $I$ variation of double sequences and investigate it's relation with $I$ and $I^*$ convergence.

Let $(X,T_X)$ and $(Y,T_Y)$ be topological spaces. A sequence $(f_n)$ of functions $f_n:X \to Y$ is graph convergent to $f:X\to Y$ if for each set $U \in T_X\times T_Y$ containing the graph $Gr(f)$ of $f$ there is an index $k$ such that $Gr(f_n) \subset U$ for $n >> k$. It is proved that if $(X,T_X)$ is a $T_1$ space, then the graph convergence implies the pointwise convergence. Moreover the uniform and graph convergences are compared, and the graph limits of sequences of continuous (quasicontinuous, cliquish, almost continuous or Darboux) functions are investigated.

We prove the generalized stability of the cubic type functional equation $$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)$$ and another functional equation $$f(ax+y)+f(x+ay)=(a+1)(a-1)^{2}[f(x)+f(y)] +a(a+1)f(x+y),$$ where $a$ is an integer with $a \neq 0, \pm 1$ in the framework of non-Archimedean normed spaces.

Let $0<p<1,\, p\neq 1/2 ,$ and let $I\subset \mathbb{R}$ be an interval. We say that the function $f:I\to \mathbb{R}$ has property ${(\rm M_p )}$ if, whenever $x,y \in I$ and $f(x)\neq f(y),$ then $f\big( px+(1-p)y\big)=f\big( (1-p)x+py\big)$. We prove that \begin{enumerate} \item[(i)] if $f$ has property ${(\rm M_p )}$ and has a point of continuity in $I,$ then $f$ is constant apart from a countable set; \item[(ii)] if $f$ is measurable and has property $({\rm M_p })$, then $f$ is constant a.e. \end{enumerate} As a corollary we obtain that if $f$ is a derivative and has property ${\rm ( M_p )}$, then $f$ is constant. Then we apply this result to solve a functional equation that appears in a variant of the Matkowski-Sut\^{o} problem.

Given an involution $f$ on $(0,\infty ),$ we prove that the set $\mathcal {C} (f) : = $ {$\lambda > 0 : \lambda f $ is an an involution} is a closed \multiplicative subgroup of $(0, \infty)$ and therefore $\mathcal{C}(F)$ is $\{1\}$, $% \intervalo$ or $\lambda ^{\mathbb{Z}}=\{\lambda ^{n}:n\in \mathbb{Z}\}$ for some $\lambda >0$, $\lambda \neq 1$. Moreover, we provide examples of involutions possessing each one of the above types as the set $\mathcal {C} (f) $ and prove that the unique involutions $f$ such that $\mathcal {C} (f) = (0,\infty)$ are $f(x)=\frac{c}{x},$ $c>0$.

A regulated function on the real line is a real valued function whose left-hand and right-hand limits exist at all points. In this paper we examine a generalization of regulated functions to functions defined on Davison Spaces, which are topological spaces with a little extra structure. Properties of such functions are discussed. Our main result concerns the set of discontinuities of these functions. We also prove that regulated functions defined on the natural numbers, with the cofinite topology, coincide with convergent sequences.

Our purpose is to study a generalized Stieltjes integral defined on a class of subsets of a closed number interval. We extend the results of previous work by the first author. Among other results, we prove that

If $M \subseteq [a,b]$ and $f$ and $g$ are functions with domain $M$ such that $f$ is $g$-integrable over $M$, and there exist left (right) extensions $f^*$ and $g^*$ of $f$ and $g$ to $[a,b]$, respectively, then $f^*$ is $g^*$- integrable on $[a,b]$ and $$ \int_a^b f^*dg^*= \int_M fdg $$

[(a)] $F$ is $G$-integrable on $[a,b]$,

[(b)] $\overline{M} \subseteq [a,b]$, and $a,b \in M$ \item

[(c)] if $z$ belongs to $[a,b] - M$ and $\epsilon$ is a positive number, then there is an open interval $s$ containing $z$ such that \break $|F(x) - F(z)||G(v) - G(u)| <\epsilon$ where each of $u$, $v$, and $x$ is in $s \cap [a,b]$, $u < z < v$, and $u \le x \le v$.

Then $F$ is $G$-integrable on $M$, and $\int_a^b FdG = \int\limits_{M}FdG$.

In this paper we consider the idea of $I$ - convergence of nets in a topological space and derive several basic properties. This idea extends the concept of $I$ - convergence of sequences considered so far in various spaces.

In [6], an infinite dimensional curve is constructed which is fairly smooth near an accumulation point of its graph, but has a null tangent set near the accumulation point. We construct extremely smooth curves which still yield such an anomalous tangency behavior.

In this paper a generalized concept of continuous multifunctions has been studied. The main goal of this paper is to study some properties concerning a new type of multifunction along with its selections.

The concept of the GAP-integral was introduced by the authors [7]. In this paper we characterize the Variational integral by the GAP-integral and present some significant convergence theorems for the GAP-integral.

Let $X$ be a connected separable linear order, a connected separable metric space, or a connected, locally connected complete metric space. We show that every continuous function $f:X\to\mathbb R$ with the property that every $x\in X$ is a local maximum or minimum of $f$ is in fact constant. We provide an example of a compact connected linear order $X$ and a continuous function $f:X\to\mathbb R$ that is not constant and yet every point of $X$ is a local minimum or maximum of $f$.

In this short paper an integral inequality posed in the $11^{th}$ International Mathematical Competition for University Students is further generalized.

We show that all measure based and all category based lower density topologies on $\mathbb{R}$ are maximally resolvable in $\mathbf{ZFC}$ and extraresolvable under $\mathbf{ZFC + MA}$. It is also noted that both maximal resolvability and extraresolvability are feeble topological properties