Let $f:\mathbb{R} \rightarrow (0,\infty)$ be a differentiable function such that $f'(x)=f(f(x))$ for all $x \in \mathbb{R}$. Prove that such a function can't exist.
All that I have found so far is that the functions $f$ and $f'$ are strictly increasing on $\mathbb{R}$. Also that $f''(x)>0$ and thus $f$ is strictly convex on $\mathbb{R}$. How do I proceed or exploit this monotonicity ?
