$\mathsf Q$-linear functions, functions with dense graph, and everywhere surjectivity

Abstract

Let $L$, $S$ and $D$ denote, respectively, the set of $\mathsf{Q}$-linear functions, the set of everywhere surjective functions and the set of dense-graph functions on $\mathsf{R}$. In this note, we show that the sets $D\setminus(S \cup L)$, $S \setminus L$, $S \cap L$ and $D\cap L \setminus S$ are lineable. Moreover, all these sets contain (omitting zero) a vector space of the biggest possible dimension, $2^c$.