On some modification of Darboux property (Q2810988)

A function \(f\:\mathbb R\to \mathbb R\) has the Darboux property if on each interval \((a,b)\subset \mathbb R\) function \(f\) assumes every real value between \(f(a)\) and \(f(b)\). \textit{A. Maliszewski} [Zesz. Nauk. Politech. Łódz. 719, Mat. 27, 87--93 (1995; Zbl 0885.26002)] investigated a class of functions which possess some stronger property. A function \(f\:\mathbb R\to \mathbb R\) has the strong Świątkowski property if for each interval \((a,b)\subset \mathbb R\) and for each \(y\) between \(f(a)\) and \(f(b)\) there exists a point \(x_0\in (a,b)\) such that \(f(x_0)=y\) and \(f\) is continuous at \(x_0\). \textit{Z. Grande} [Colloq. Math. 117, No. 1, 95--104 (2009; Zbl 1177.26005)] considered some modification of strong Świątkowski property changing the continuity with approximate continuity. In this paper, the authors introduce some family of functions \(f\:\mathbb R\to \mathbb R\) modifying the Darboux property analogously to as it was done by Grande [loc. cit.], and changing approximate continuity with \(\mathcal {I}\)-approximate continuity, i.e., continuity with respect to the \(\mathcal {I}\)-density topology. They prove that their family is a strongly porous set in the space of Darboux functions having the Baire property and that each function from their family is quasi-continuous.