Calculus help? Can I say this function is a polynomial?

Question

I have a function $f(x)$ and $g(x) = f(f(x))$ both from domain D to range D.

$f$ and $g$ are bijective. Furthermore, $f$ is increasing and bounded above by a polynomial.

I think I read that if there is some $n$ where $g^{(n)}(x) = 0$ for all $x$, then $g$ is a polynomial.

Is this enough to assume $f$ is a polynomial?

Answer 1

Try $f(x) = x^{\sqrt{2}}$ with $D = (0,1)$.