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 . Since 1875, when Darboux functions were defined, many papers have appeared about their properties. Proofs of many properties of real Darboux functions of real variables are very important, because the family of Darboux functions includes many important classes of functions; for example, continuous functions and functionally connected functions. (See the next page for the definition.) Mutual inclusions among several families of Darboux-like functions inspired questions concerning the size of particular sets in spaces of functions. One of the questions is how strong the inclusions are. Similar issues have already been considered in a great number of papers such as [4], [5] and [6]. In [3] Jȩdrzejewski noticed that each continuous function is functionally connected and each functionally connected function is a Darboux function. It is not difficult to show that there exists a discontinuous, functionally connected function and there exists a Darboux function which is not functionally connected. Moreover, it turns out that in the space of bounded Darboux functions (with the metric of uniform convergence) the set of bounded functionally connected functions is boundary (its complement is a dense set) and in the space of bounded functionally connected functions the set of bounded upper (lower)