Construct a diffeomorphism that extends the identity map Is there a diffeomorphism  $f:[1,+ \infty) \rightarrow [1,3)$, such that $f$ restricted on $[1,2]$ is the identity map? I think its true, but I don't know how to construct one.

It suffices to find an increasing diffeomorphism $g : (2, \infty) \to (2,3)$ such that $\lim \limits _{ \begin {array} {c} x \to 2 \\ x>2 \end{array} } g'(x) = 1$ (this last condition will allow you to glue $g$ to the identity while maintaining a continuous derivative of the resulting function). There aren't many functions with a simple explicit description that squeeze an interval of infinite length into one of finite length; the usual suspects are $\arctan, \Bbb e^{-x}, \frac {2x} {1+x^2}$... You'll also have to compose with a rescaling and a translation in order to land in $(2,3)$.  The choice $g(x) = 3 - \Bbb e ^{2-x}$ does the job.
Therefore, $f(x) = \left\{ \begin{array} {ll} x, && x \in [1,2] \\ 3 -  \Bbb e ^{2-x}, && x \in (2, \infty) \end{array} \right.$ is the desired diffeomorphism. (Note that it is not twice-derivable, though, i.e. it is a diffeomorphism as in real analysis, but not as in differential geometry.)
