Is the identity map a diffeomorphism? Unfortunately, googling this question leads to conflicting answers. According to  this source, the identity map on any smooth manifold is a diffeomorphism,
 but it's not according to this.   I appreciate it if someone gave a definitive answer.
 A: Let us fix a smooth manifold $M$.


*

*is the identity map $i:M\to M$ smooth?

*is it bijective? 

*what is the inverse function?

*is the inverse function bijective?
Can you answer these questions?
As for your reference: the book does not say that the identity map of a smooth manifold is not a diffeo: it gives an example to show that if $M$ and $M'$ and two smooth manifolds on the same topological space, then the identity function $M\to M'$ is not necessarily smooth. This is a claim rather rather different  to «the identity map of a smooth manifold is not smooth».
A: Yes, the identity map is a diffeomorphism, and the derivative at any point $p$ is just the identity on $T_pM$. Maybe it is best to see this in terms of directional derivatives. Write $I$ for the identity map. Fix a curve $\phi(t): \mathbb{R} \to M$ with $\phi'(t) = X$ for some $X \in T_pM$ and then compose with the identity map. Then $D_X I = (I\circ \phi)'(t) = \phi'(t) = X$.
Regarding your second reference, the author there is giving an example of two $C^{\infty}$ structures on $\mathbb{R}$ that are different. The issue you are having is that the ``identity map'' there really takes $\mathbb{R}$ with one smooth structure to $\mathbb{R}$ with another smooth structure. So it doesn't have to be smooth!
But if you consider the identity map on a manifold $M$ (with fixed smooth structure-  if someone utters the words "smooth manifold" then they mean a topological manifold together with a smooth structure so that is embedded in the definition) then the identity map is always a diffeomorphism.
A: I think it does not necessarily a diffeomorphism if you think identity map as a map from a set to itself. However if you consider identity map from a smooth manifold to itself (i.e to same set with same smooth structure) then it is a diffeomorphism. Here the point is this, differentiability related with your smooth structure and you can put different smooth structures on the same set. 
