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 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 ℝ → ℝ, every non-zero element of which is everywhere surjective. © 2004 American Mathematical Society.