I am reading a text stating:
“It is easy to give examples to show that a $C^{\infty}$ homeomorphism need not be a diffeomorphism. For any integer $n > 1$ the map $x^{2n-1}: \mathbb R \to \mathbb R$ is such an example, since the inverse does not have a derivative at $0$. If we change our viewpoint and consider the $C^{\infty}$ homeomorphism as the identity map on an underlying topological manifold, we get examples of different $C^{\infty}$ structures on the same space, although the resulting manifolds may be diffeomorphic under another map. The example given then shows that if we take $\{x^{2n-1}\}$ as a basis for a $C^{\infty}$ structure on $\mathbb R$ we get different structures for different $n$’s.”
How is it possible to construct different smooth structures using the identity map? Also apparently every function $x^{2n-1}$ gives rise to a smooth structure, how?