Dense and σ-porous subsets in some families of cliquish functions

Abstract

A function f: ℝ → ℝ is internally cliquish (internally quasi-continuous) if it is cliquish (quasi-continuous) and the set of discontinuity points of f is nowhere dense. We prove that the set of internally cliquish (internally quasi-continuous) functions is dense and σ-strongly porous set in the family of cliquish (quasi-continuous) functions. The analogous result is obtained also for the family of functions having the Świątkowski property.