This may be related to solving $f(f(x))=g(x)$. Let $C(\mathbb{R})$ be the linear space of all continuous functions from $\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \exists\; f\in C(\mathbb{R})$ s.t. $f\circ f=g \}$ . Is there some infinite dimensional (or, at least, bidimensional) linear subspace of $C(\mathbb{R})$ contained in $\mathcal{S}$ ?

P.S. As a remark, there is a [maybe] interesting connection between How to solve $f(f(x)) = \cos(x)$? and Borsuk pairs of Banach spaces . Namely, let $E$ be the closed subspace of $C[-1,1]$ consisting of all even functions, and let $K$ be the closed unit ball of $E$. Then the continuous mapping $\Psi:$ $K$ $\rightarrow$ $E$ expressed by $\Psi(f)$ $:=$ $f\circ f$ $+$ $\left(\left\Vert f\right\Vert _{\infty}-1\right)\cdot\cos$

is odd on $\partial K$, and has no zeros in $K$.