Let
$$\mathbb{R}=\{(\mathbb{R},\phi)\}\quad \phi(x)=x$$ $$\mathbb{R}'=\{(\mathbb{R},\psi)\}\quad \psi(x)=x^{1/3}$$
be two atlases for the real line. Show that there is a diffeomorphism between $\mathbb{R}$ and $\mathbb{R}'$.
There is a theorem that says that $\phi^{-1}\circ \psi = x^{1/3}$ and $\psi^{-1}\circ \phi = x^{3}$ are diffeomorphisms.
What confuses me is that the exercise gives a "Hint" saying, "the identity map is no a diffeomorphism since it is not smooth". I don't understand the hint, even though I think my answer is correct. Can somebody explain why $id(x):\mathbb{R}\to\mathbb{R}, id(x)=x$ is not an infinitely differentiable function?