On some Modification of Świtąkowski Property

We introduce some families of functions ƒ : ℝ → ℝ modifying the Darboux property analogously as it was done by [Maliszewski, A.: On the limits of strong Świątkowski functions, Zeszyty Nauk. Politech. Łódź. Mat. 27 (1995), 87-93], replacing continuity with A-continuity, i.e., the continuity with respect to some family A of subsets in the domain. We prove that if A has (*)-property then the family DA of functions having A-Darboux property is contained and dense in the family DQ of Darboux quasi-continuous functions.