Opuscula Math. 34, no. 4 (2014), 777-788
http://dx.doi.org/10.7494/OpMath.2014.34.4.777

Opuscula Mathematica

On some subclasses of the family of Darboux Baire 1 functions

Gertruda Ivanova
Elżbieta Wagner-Bojakowska

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.

Keywords: Darboux property, strong Świątkowski property, Baire property, $$\mathcal{I}$$-approximate continuity, quasi-continuity.

Mathematics Subject Classification: 26A15, 54C08.

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