On Small Subsets of the Space of Darboux Functions

Abstract

We prove that the set F of all bounded functionally connected functions is boundary in the space of all bounded Darboux functions (with the metric of uniform convergence). Next we prove that the set of bounded upper (lower)semi-continuous Darboux functions and the set of all bounded quasi-continuous functionally connected functions is porous at each point of the space F.

On Strong Porosity of Some Families of Functions

“…The investigations focused mainly on metric spaces. There are many results discussing the comparison of porosity for different subsets of the function spaces (see [7][8][9][10]). It is interesting to consider not only continuous functions with respect to a given topology but also continuous with respect to the families which do not form a topology (see [5,11,12]).…”

Porosity in Spaces of Darboux-Like Functions

“…The porosity of special sets in spaces of Darboux-like functions has been studied, for example, in [10] and [11]. For functions f : R → R, it is known that C ⊂ Ext ⊂ AC ⊂ Conn ⊂ D ⊂ PC [12].…”