Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$

Question

Find all continuous functions $f:\mathbb R\to\mathbb R$ such that $f(f(x))=e^{x}$

I didn't solve this problem, but I proved that f is increasing and $ x<f(x)<e^{x} $

please help

Answer 1

You may wish to have a look at Kneser's paper Reelle analytische Losungen der Gleichung $\varphi(\varphi(x)) = e^x$ und verwandter Funktionalgleichungen.