Cardinal invariants connected with adding real functions

Abstract

In this paper we consider a cardinal invariant related to adding real functions defined on the real line. Let \(\mathcal{F}\) be such a family, we consider the smallest cardinality of a family \(\mathcal{G}\) of functions such that \(h+\mathcal{G}\) has non-empty intersection with \(\mathcal{F}\) for every function \(h\). We note that this cardinal is the additivity, a cardinal previously studied, of the compliment of \(\mathcal{F}\). Thus, we calculate the additivities of the compliments of various families of functions including the Darboux, almost continuous, extendable and perfect road functions. We briefly consider the general relationship between the additivity of a family and its compliment.