On quasi-uniform convergence of sequences of s₁-strongly quasi-continuous functions on Rⁿ

Abstract

A function $f: \mathbb{R}^m \to \mathbb{R}$ is called $s_1$-strongly quasi-continuous at a point $\mathbf{x}\in \mathbb{R}^m$ if for each real $\varepsilon >0$ and for each set $A\ni {\bf x}$ belonging to the density topology, there is a nonempty open set $V$ such that $\emptyset \neq A \cap V \subset f^{-1}((f(\mathbf {x})- \varepsilon ,f(\mathbf {x})+ \varepsilon ))\cap C(f),$ denotes the set of continuity points of $f$. It is proved that every $\lambda $-almost everywhere continuous function $f:\mathbb{R}^m \to \mathbb{R}$ is the quasi-uniform limit of a sequence of $s_1 $-strongly quasi-continuous functions and that each measurable function $f:\mathbb{R}^m \to \mathbb{R}$ is the quasi-uniform limit of a sequence of approximately quasi-continuous functions $f:\mathbb{R}^m \to \mathbb{R}.$