Lineability and K-linear discontinuous functions

Abstract

It is well known that there are Q-linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory.Generalizations of classical results of Q-linear discontinuous functions are also analyzed.