On some subclasses of the family of Darboux Baire 1 functions

Abstract

Abstract. We introduce a subclass of the family of Darboux Baire 1 functions f : R → 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 I-approximate continuity, i.e. continuity with respect to the 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 DB1 of Darboux Baire 1 functions, equipped with the supremum metric.

On the pointwise limits of sequences of Świątkowski functions

“…Thus the condition S(R, A, 0) can be treated as a generalization of the Świątkowski property related to a fixed set A. An analogous modification of the strong Świątkowski property has been considered by Marciniak and Szczuka [15], see also [6].…”

Comparison of Some Subfamilies of Functions Having The Baire Property

“…The notion of porosity in spaces of Darboux-like functions was studied among others by J. K u c n e r, R. P a w l a k, B.Ś w ia t e k in [8], by H. R o s e n in [13] and by G. I v a n o v a, E. W a g n e r-B o j a k o w s k a in [6].…”