When a local diffeomorphism is a diffeomorphism

Question

Why is it the case that a local diffeomorphism $f:\mathbb{R}\rightarrow\mathbb{R}$ is a diffeomorphism if $f$ is injective?

Answer 1

Since $f$ is injective, $f^{-1}$ exists. And since $f$ is a local diffeomorphim, $f^{-1}$ is also differentiable. Therefore, $f$ is a diffeomorphism.

Answer 2

Such a map is not a diffeomorphism. A counter-example would be the identity map

$$id:[0, 1] \rightarrow \mathbb{R}, \quad x \mapsto x.$$

Such a map is obviously a local diffeomorphism, but it certainly is not a diffeomorphism, or even a homeomorphism, as it fails the surjectivity condition.

What I think the post wants is a smooth embedding, which is a diffeomorphism onto its image. Hence, if $f:X \rightarrow Y$ is an injective local diffeomorphism, then we can find an inverse map $$f^{-1}:f(X) \rightarrow X, \quad y \mapsto f^{-1}(y),$$

which is a function because $f$ is injective. It is also smooth because $f$ is a lcoal diffeomorphism. Therefore, the function

$$f:X \rightarrow f(X)$$

is a diffeomorphism because $f$ has a smooth inverse.