Disjoint continuous images with values in the inductive boundaries

Abstract

Generalizations of the classical results of Dini and Osgood on sequences of continuous functions are obtained. Based on these generalizations, we establish a Bairetype theorem concerning the size of their point set of joint continuity of separably continuous mappings of products of Baire spaces and spaces with first countability axiom in certain inductive limits of increasing sequences of locally convex metrizable spaces containing, in particular, such well-known nonmetrizable spaces as the space of finitary sequences and space of Schwartz sampling functions.