Lineability and additivity in R(R).

Abstract

We give a condition for a family of functions to be lineable by means of its additivity. The novelty presented here is that we solve the lineability problem using a technique that is not constructive, as are most approaches to this problem. We relate the notions of additivity and lineability and use this relation to give a general method to find the lineability of large families of functions. We also study more examples of pathologically behaving functions, in particular, the class of Jones functions, which is a highly pathological subclass of perfectly everywhere surjective functions. We work on the additivity, lineability, and main properties of this class.