Lineability and spaceability of sets of functions on $\mathbb {R}$

Abstract

We show that there is an infinite-dimensional vector space of differentiable functions on $\mathbb {R},$ every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension $2^c$ of functions $\mathbb {R} \to \mathbb {R},$ every non-zero element of which is everywhere surjective.