ca.analysis and odes - $fcirc f=g$ revisited

This may be related to solving f(f(x))=g(x). Let
$C(mathbb{R})$ be the linear space of all continuous functions from
reals to reals, and let $mathcal{S}$ $:=$ { $gin C(mathbb{R})$ $;$ $exists$ $fin C(mathbb{R})$ s.t. $fcirc f=g$ } . 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 $Psi(f)$ $:=$ $fcirc f$ $+$ $left(leftVert frightVert _{infty}-1right)cdotcos$

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