A generalization of Darboux-Froda theorem and its applications

In real analysis, the Darboux-Froda theorem states that all discontinuities of a real-valued monotone functions of a real variable are at most countable. In this paper, we extend this theorem to a family of monotone real vector-valued functions of a real variable arising from dynamical systems. To this end, we explore some essential characteristics of countable and uncountable sets by the notions of strong cluster points, upper and lower strong cluster points, and establish the existence of strong cluster point sets, upper and lower strong cluster point sets for an uncountable set. With the help of these strong cluster point sets, we establish a jump lemma that helps characterize the discontinuities of the family of monotone vector-functions. Then we introduce the notion of distinction set and prove the existence of a distinction set. Making use of the upper and lower strong cluster points of the distinction set and the jump lemma, we prove the Darboux-Froda extension theorem. Moreover, we also present two applications of the generalized Darboux-Froda theorem.