On Density and Sigma-Porosity in Some Families of Darboux Functions

Abstract

A cliquish function $f: \mathbb R\to\mathbb R$ is internally cliquish if its set of discontinuity points is nowhere dense. A Baire $1$ function $f: \mathbb R\to\mathbb R$ is internally Baire $1$ if its set of discontinuity points is nowhere dense. In this paper we proved that the set of Darboux internally cliquish (Baire $1$) functions is dense and $\sigma$-strongly porous in the family of Darboux cliquish (Baire $1$) functions.