In this paper we investigate properties of some rings and ideals of real functions. Moreover we present some generalizations of the notion of density point.

Let $X$ be a separable metrisable Baire space without isolated points. It is proven that every cliquish function $f:X\to R$ is the sum of four quasi continuous functions.

We give an example of a sup-measurable and non-measurable function in the general case associated with products of $\sigma$-ideals. We show that any behaviour of a function $F:{\mathbf{R}}^2\to \mathbf{R}$ with respect to $(L)$- and $(B)$-measurability, and $(L)$- and $(B)$-sup-measurability is possible. Continuum Hypothesis is replaced by weaker conditions. Finally, the notion of quasi-sup-measurability is considered.

By similar methods as Calderón [1] and Stein [10], we prove some restriction theorems on weighted potential spaces when the weight satisfies Muckenhoupt’s $A_p$ condition. We then construct potential spaces of mixed norm and find that mixed norm potential spaces are indeed a natural object in the study of restriction theorems. With the help of mixed norm spaces, we obtain a better version of restriction theorem.

Suppose $a\ne b$ are non-zero. We prove that if either $\lim {\inf }_{t\to 0-}(f(x+bt)-f(x+at))\ge 0$ or $\lim {\inf }_{t\to 0+}(f(x+bt)-f(x+at))\ge 0$ holds at each point of a set $E\subset \mathbf{R}$ which has the Baire property (is measurable), then $f$ is continuous at each point of $E$ apart from a first category set (a set of measure zero). We prove this via a generalization of the Uher-Thomson symmetric covering theorem.

A combinatorial characterization of measurable filters on a countable set is found. We apply it to the problem of measurability of the intersection of nonmeasurable filters.

Given two Borel measures $\mu$, $\nu$ in ${\mathbf{R}}^1$, finite on compacts, we deal with the conditions $v\in A_{\infty }(\mu )$ and $\nu \in A_{\infty }^{+}\left(\mu \right)$. We point out several characterizing statements, expressed in terms of maximal functions, $A_p$ classes, reverse Hölder inequalities, exponential type conditions, and their symmetric versions. We use ideas from [1] and [19] and extend their results to this rather general context (in the one-sided case we assume that $\mu$ and $\nu$ are mutually absolutely continuous). We consider an application to the Gehring lemma.

The concept of convergence of real sequences is completely characterized by six properties, each essentially a theorem from the theory of limits of sequences of real numbers. Consequently, the foundation of the calculus can be constructed without the use of the “$ϵ,\delta$” definitions of Cauchy. The six properties are versions of: (1) the scalar multiple of a convergent sequence is convergent; (2) the corresponding sums of equivalent convergent sequences are equivalent; (3) the Squeezing Theorem; (4) all subsequences of a convergent sequence converge and are equivalent; (5) the sequence $1,-1,1,-1,\cdots$ does not converge; and (6) every divergent and bounded sequence has two non-equivalent, convergent subsequences. Five of the six properties are shown to be necessary for the characterization of convergence. The question of the independence of the Squeezing Theorem from the other five properties remains unresolved.

It is observed that the Walsh set $E\subseteq X\times Y$ which has property $(s^0)$ and is nonmeasurable can satisfy the following conditions simultaneously: $E$ is nonmeasurable in some strong sense; $E$ is the graph of a bijection; $E$ meets each function of a fixed family of size $c$, in a set of size .

It is shown that there exist Cantor subsets $X$ of ${ℝ}^N$ and bi-Lipschitz maps $f:X\to f(X)\subset \mathcal{H}$, where $\mathcal{H}$ is an infinite dimensional Hilbert space, such that $f$ is not strongly differentiable at any point of $X$. Furthermore, for each such $X$ and $f$ the image $M=f(X)$ has the property that for any $N\ge 1$ and any differentiable map $F:{[0,1]}^N\to \mathcal{H}$ with $dF(x)$ nonsingular for all $x\in {[0,1]}^N$, the set $F^{-1}(M)$ is a finite set. Hence, $f$ can agree with a nonsingular differentiable map at most on a finite set.