Methods of comparison of families of real functions in porosity terms

Abstract

We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.

On Topologies Generated by Lower Porosity

“…The main idea is that we modify the "ball" definition of nowhere density by the request that the sizes of holes should be estimated. Usually, the notion of the (upper) porosity of sets is used in many aspects, see for example [4][5][6]10,12,13]. We deal with the lower porosity, which also be considered in some papers, [11,12].…”

“…In [1] J. Borsík and J. Holos defined families of porouscontinuous functions f : R → R. Some properties of porouscontinuity can be found in [1,2,10]. Applying their ideas and replacing standard porosity in R by the lower porosity in X we transfer this concept for real functions defined on (X, ).…”

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]).…”

Lower Porosity on R2

“…Recently, the concept of porosity is used not only to study subsets of a metric space but also to compare families of functions. By applying the notion of porosity one can describe how "small" one class of function in the other is, (see [2][3][4]). In the study of these comparison of classes of functions, it was found that it was possible to strengthen results by using the lower porosity instead of the (upper) porosity.…”