All of the maps
$$ F(x) = x^4 \\ G(x) = \exp (\exp x) \\ H(x) = \sin (\sin x) $$
can be expressed as the self-compositions of the functions
$$ f(x) = x^2 \\ g(x) = \exp x \\ h(x) = \sin x $$
So this led me naturally to the question whether other functions can be expressed as the self composition of another function. So this led me naturally to the question:
Does there exist a continous function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x \in \mathbb{R}$, $f(f(x)) = \exp (x)$?
