Maximal lineability of the set of continuous surjections

Abstract

Let $m,n$ be positive integers. In this short note we prove that the set of all continuous and surjective functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ contains (excluding the $0$ function) a $\mathfrak{c}$-dimensional vector space. This result is optimal in terms of dimension.