On the set of functions possessing the property (top) in the space of Darboux and Światkowski functions (Q1332584)

We say that a Darboux and Światkowski function \(f: \mathbb{R}\to \mathbb{R}\) has the property (top) if there exists a topology \(T\) stronger than the natural topology in \(\mathbb{R}\) such that the ring \(C(T)\) of all continuous functions in this topology consists of Darboux and Światkowski functions, and \(f\in C(T)\). The main result of the paper is included in the following theorem: The family of functions having the property (top) is porous in the space of bounded Darboux and Światkowski functions with the metric of uniform convergence.