Uniform Limits of Preponderantly Continuous Functions

Abstract

There are a few nonequivalent definitions of preponderant continuity \cite{bru1,den1,gra,kow,oma}. In this paper we investigate uniform limits of preponderantly continuous functions. We show that preponderantly continuous functions in the O’Malley sense are closed under uniform limits, while the family of preponderantly continuous functions in the Denjoy sense are not closed under uniform limits and find closure of this family in the topology of uniform convergence. Finally, we prove that the set of preponderantly continuous in Denjoy sense functions is a first category subset of its uniform closure.