On some subclasses of the family of Darboux Baire 1 functions

Abstract. We introduce a subclass of the family of Darboux Baire 1 functions \(f:\mathbb{R}\rightarrow\mathbb{R}\) modifying the Darboux property analogously as it was done by Z. Grande in [On a subclass of the family of Darboux functions, Colloq. Math. 17 (2009), 95-104], and replacing approximate continuity with \(\mathcal{I}\)-approximate continuity, i.e. continuity with respect to the \(\mathcal{I}\)-density topology. We prove that the family of all Darboux quasi-continuous functions from the first Baire class is a strongly porous set in the space \(\mathcal{DB}_1\) of Darboux Baire 1 functions, equipped with the supremum metric.