Inverse of a differentiable function equal to its derivative then f is analytic

Question

I've found a nice problem concerning analytic functions. Here it is:

Let $f: (0, \infty) \rightarrow \mathbb{R}$ be a function differentiable on $(0, \infty)$ and such that $f^{-1} = f'$. Prove that $f$ is analytic on $(0, \infty)$.

I'm not sure if it's relevant, but I know that $f$ cannot be a bijection :)

Could you help me?

Answer 1

The answer is affirmative, as proven in this MathOverflow post. The answerer, José Hdz. Stgo., assumes $f:[0,\infty) \to [0, \infty)$ and $f(0) = 0$, but these additional hypotheses don't affect the argument regarding analyticity.