Are homeomorphic differentiable manifolds actually diffeomorphic? Let $M$ and $N$ be two n-dimensional smooth manifolds.Suppose their underlying topological spaces are homeomorphic through $f$. Does $f$ automatically become a diffeomorphism with respect to the given smooth structures? If not, can I adjust any of the smooth structures to make $f$ a diffeomorphism? What if I restrict the manifolds to be embedded manifolds in Euclidean space $\mathbb{E}^n$ with endowed topology and smooth structure?

Let $M=N=\mathbb R$, endowed with its usual structure of a smooth manifold, and let$f:M\to N$ be the map such that $f(x)=x^3$. This is a homeomorphism but not a diffeo!
Now, if $f:M\to N$ is a homeomorphism of smooth manifolds, you can always «adjust» the smooth structures so that $f$ becomes a diffeo: indeed, you should be able to prove the following:

if $M$ is a smooth manifold, $N$ a topological space and $f:M\to N$ a homeomorphism, then there is a structure of smooth manifold on $N$ such that $f$ becomes a diffeomorphism.

