show $\alpha^{-1}$ is continuous if $\alpha \in C^r$ is bijective, and $D\alpha$ is nonsingular

Question

In the book of Analysis on Manifolds by Munkres, at page 196, it is given that

However, if $\alpha$ is of class $C^r$, one-to-one and onto, and $D\alpha$ is nonsingular for $x \in U$, isn' $\alpha^{-1}$ continuous by the inverse Function Theorem? If so, why does the author include requirement $(2)$? Is it for pedagogical purposes that for the case with boundary, we cannot use the Inverse Function Theorem?

Note that, for manifolds with boundary, the space $\mathbb{R}^n$ can be $\mathbb{H}^k$, too, and for this particular case, I have already seen a counterexample where $\alpha^{-1}$ is not continuous (see page 202 in the same book).

Edit:

I think I figured what I was missing; Can you check the logic that I have given in the comments of the given answer.

Answer 1

Well,... it turned out that the inverse function theorem is valid only for map from $\mathbb{R}^n $ to $\mathbb{R}^n $, which is not the case in here, so we cannot use it.

Answer 2

If you read two pages down, i.e. page 198, there is an example, namely

$$\alpha: t \mapsto \sin2t(\vert \cos t \vert, \sin t)$$ defined from $(0,\pi)$ into $\mathbb R^2$ which explains why the hypothesis $\alpha^{-1}$ continuous is important. The Implicit Function Theorem doesn’t work as we’re working on manifolds not in full space. The example gives a clear highly on that: the image of the open subset $(0,\pi)$ is not an open subset of $\mathbb R^2$.