On quasi-uniform convergence of a sequence of s. q. c. functions (Q2702775)

In [\textit{Z. Grande}, Math. Slovaca 44, No. 3, 297-301 (1994; Zbl 0837.26003)] it is proved that every real cliquish (i.e., the set \(D(f)\) of all discontinuity points of \(f\) is of the first category) function \(f: \mathbb R\to \mathbb R\) is the quasiuniform limit of a sequence of Darboux quasi-continuous functions. In this paper it is shown an analogous result for the measure case: every almost continuous (i.e., \(D(f)\) is of measure zero) function \(f: \mathbb R\to \mathbb R\) is the quasi-uniform limit of a sequence of Darboux strongly quasi-continuous functions. (A function \(f: \mathbb R\to \mathbb R\) is said to be strongly quasi-continuous (briefly s.q.c.) at a point \(x\) if for every set \(A\) containing \(x\), which is open in the density topology, and for every positive real \(\eta \), there is an open interval \(I\) such that \(I\cap A\neq \emptyset \) and \(|f(t)-f(x)|<\eta \) for all \(t\in I\cap A\)).