An error in Section 4 of \cite{NS} is corrected.

Some generalizations of the notions of approximate quasicontinuity on $\mathbb{R}^m $ and the maximal families (additive, multiplicative, lattice and with respect to the composition) for these classes of functions are investigated

In this note we study descriptive set theoretic as well as measure theoretic properties of smooth images of the irrationals.

The system of functional differential equations (1) has a continuously differentiable solution for every value of the parameter $a$. The boundary values and $a$ are related with $d(2-a)=c(2+a)$. When $a\in S$ where $$S=\left\{ 2^{2n+1}:n=1,2,3,\ldots\right\},$$ the system (1) has infinitely many solutions with boundary values $c=0$ and $d=0$. For all other values of $a$, the system \eqref{equation1_1} has a unique solution. \begin{equation} \tag{$1$}\label{equation1_1} \left \{ \begin{array} {l l } F^{\prime}(x)=a F(2x) & \mathrm{ if} \: 0\leq x\leq \dfrac{1}{2} \\ F^{\prime}(x)=a F(2-2x) & \mathrm{ if} \: \dfrac{1}{2} \leq x\leq 1 \\ F(0)=c, F(1)=d. & \end{array} \right. \end{equation}

Under the assumption of projective determinacy, we show that one can obtain non-$\sigma$-porous compact subsets for given projective non-$\sigma$-porous sets, with respect to the regular and strong porosities for dimension zero compact metric spaces.

We prove that a purely unrectifiable self-similar set of finite 1-dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point.

We deal with the class of functions defined as a sum of a uniformly convergent series of functions continuous both on a closed set and on its complement. Such functions are mentioned in the literature, e.g., [1], [2], [3], [4].$ We investigate the particular class of derivatives.

We consider self-similar Borel probability measures $\mu $ on a self-similar set $E$ with strong separation property. We prove that for any such measure $\mu $ the derivative of its distribution function $F(x)$ is infinite for $\mu $-a.e. $x\in E$, and so the set of points at which $F(x)$ has no derivative, finite or infinite is of $\mu $-zero.

We discuss and compare some generalized types of continuity and differentiability. In particular, we focus on a notion of differentiability based on an integral average, and we establish a link with an approximation procedure by polynomials.

A. Jonsson has constructed wavelets of higher order on self-similar sets, and characterized Besov spaces on totally disconnected self-similar sets, by means of the magnitude of the coefficients in the wavelet expansion of the function. For a class of self-similar sets, W. Jin shows that such wavelets can be constructed by recursively calculating moments. We extend their results to a class of graph-directed self-similar sets, introduced by R. D. Mauldin and S. C. Williams.

Using probabilistic methods, we find the exact Hausdorff measure function and dimension of sets of dyadic Lipschitz points (i.e., slow points) for functions belonging to particular Zygmund-type classes. We then explore, in depth, the relationship between sets of slow points and sets of standard Lipschitz points, both in the particular case of the van der Waerden--Takagi function and for more general dyadic Zygmund functions.

Let $f$ be a function in $C([0,1])$. We denote by $f_p$ the best approximant to $f$ in $L_p([0,1])$ by nondecreasing functions. It is well known that the limit $f_{*}:=\lim_{p\to\infty}f_p$ exists and $f_{*}$ is a best approximant to $f$ in $C([0,1])$ by nondecreasing functions. In this paper we show an explicit formula for the function $f_{*}$ and we prove some additional minimization properties of $f_{*}$.

A classical theorem of Luzin is that the separation principle holds for the $\mathbf{\Pi}^0_\propto$ sets but fails for the $\mathbf{\Sigma}^0_\propto$ sets. We show that for every $\mathbf{\Sigma}^0_\propto$ set $A$ which is not $\mathbf{\Pi}^0_\propto$ there exists a $\mathbf{\Sigma}^0_\propto$ set $B$ which is disjoint from $A$ but cannot be separated from A by a $\mathbf{\Delta}^0_\propto$ set $C$. Assuming $\mathbf{Pi}^1_1$-determancy it follows from a theorem of Steel that a similar result holds for $\mathbf{Pi}^1_1$ sets. On the other hand assuming V=L there is a proper $\mathbf{Pi}^1_1$ set which is not half of a Borel inseparable pair. These results answer questions raised by F.Dashiell.

Let $(X,T_X)$ and $(Y,T_Y)$ be topological spaces and let $(Z,\rho _Z)$ be a metric space. In this article we characterize the sets of all continuity points of symmetrically cliquish functions from $X\times Y$ to $Z$ and the sets of continuity points of symmetrically quasicontinuous functions from $\mathbb{R}^2$to $\mathbb{R}$.

We present a new classification of Baire one star functions and examine sums of functions from the defined classes.

Let $f$ be a distributionally chaotic map of the interval such that the endpoints of the minimal periodic portions of any basic set are periodic. Then the principal measure of chaos, $\mu _p(f)$, is not greater than twice the spectral measure of chaos $\mu _s(f)$. This proves an assertion of Schweizer et al. in a special case.

In this paper, we will investigate the following question: Given $ C \in (0,1) $ and a sequence $ A_n \subseteq [0,1] $ with $ \lambda(A_n)=C $, when does there exist a subsequence $ A_{n_i} $ such that $ \lambda( \cap_i A_{n_i} ) >0 $? We will show that the answer to this question can be characterized by the properties of a function $g$ which will be a weak $ L^1 $ limit of characteristic functions.

We consider a generalized version ($\mathsf{GES}$) of the well-known Severini-Egoroff theorem in real analysis, first shown to be undecidable in $\mathsf{ZFC}$ by Tomasz Weiss in [6]. This independence is easily derived from suitable hypotheses on some cardinal characteristics of the continuum like $\mathfrak{b}$ and ($\mathcal{N}$)

In much the same way that curvature provides a measure of the nonlinearity of a curve, aberrancy provides a measure of the noncircularity of a curve. Curvature can be defined in several ways, but they all result in the same formula. In contrast, different approaches to aberrancy yield different formulas. We consider a number of different approaches to aberrancy and show that there are some interesting and unexpected connections between them. This is a largely unexplored concept that can be used to generate projects for students in real analysis and calculus.

Short and elementary proofs of the well-known Stirling formula for the discrete Gamma function $\Gamma(n)$ have been given by several authors. In this note, a well-known identity and Stirling's formula for the continuous Gamma function $\Gamma(x)$ are deduced in a different and short way from a simple and elementary proposition.

This paper deals with the generalization of the density points with respect to category. Topologies using this concept of density are introduced, and their properties are investigated.