Here's the question : A smooth bijective map of manifolds need not be a diffeomorphism. In fact, show that $$f:\mathbb{R^1}\rightarrow {R^1}$$ $$x\rightarrow f(x)=x^3,$$ is an example.
I would like to do this problem, but I'm really not sure I understand it. The question does mean that there are smooth bijective map of manifold without as the map is a diffeomorphism. Could someone explain to me the meaning of this question?
The important fact here is that you need to take a closer look at the inverse $f^{-1}$.
The inverse of $x^3$ is $\sqrt[3]{x}$ the differential of the inverse $\frac{d}{dx}f^{-1}=\frac{1}{3 \sqrt[3]{x^2}}$ outside of $0$ and nonexistent at $0$ therefore $f$ is no diffeomorphism since the inverse is not differentiable at $0$.