Unpublished results of K. Pekár and H. Zlonická on preponderant derivatives and \(M_ 4\)-sets (Q909811)

The paper deals with real functions of a real variable. In the first part of the paper three notions of the preponderant continuity (strong preponderant continuity, preponderant continuity and weak preponderant continuity) and three kinds of preponderant derivatives \(((s)-f'_{pr}\), \(f'_{pr}\) and \((w)-f'_{pr})\) are defined. Some basic properties of these notions are stated. E.g., if f is weakly preponderantly continuous, then f is Darboux and in Baire class 1; if f has a preponderant derivative (finite or infinite), then \(f'_{pr}\) is in Baire class 1. In the second part of the paper the following notion of a D-set is introduced: An \(F_{\sigma}\)-set M is said to be a D-set if there exists \(d>0\) such that for each \(x\in M\) there exists a set \(M_ x\subset M\) such that \(d(M_ x,x)=d\) (d(A,x) - the density of A at x). It is shown that Zahorski's \(M_ 4\) sets can be characterized as countable unions of D-sets.