On some modification of Świątkowski property

Abstract

We introduce some families of functions $f :\mathbb{R} \to \mathbb{R}$
modifying Darboux property analogously as it was done by A. Mal-
iszewski in [14], changing continuity with $\mathcal{A}$-continuity, i.e. continuity with respect to some family $\mathcal{A}$ of subsets in the domain.
We prove that if $\mathcal{A}$ has the $(*)$-property then the family $\mathcal{D_A} of
functions having $\mathcal{A}$-Darboux property is contained and dense in
the family $\mathcal{DQ}$ of Darboux quasi-continuous functions.