TITLE: Continuous bijection on open interval is homeomorphism
QUESTION [7 upvotes]: Suppose that $a,b\in\mathbb R$ with $a<b$ and that $f:(a,b)\to\mathbb R$ is continuous and bijective. I would like to prove that $f$ is a homeomorphism using elementary methods (no resort to invariance of domain, for instance).
To my surprise, I have not found a straightforward, standard (textbook) reference for this result. If you know any or you have a clever hint in mind, I would be grateful if you could share it.

REPLY [2 votes]: There is an elementary, general, textbook result which answers your question, and which relies on the concept of a proper map $f : X \to Y$, meaning that the pre-image of every compact subset of $Y$ is compact in $X$. 
Theorem: If $f : X \to Y$ is a continuous, proper bijection and $Y$ is a compactly generated Hausdorff space then $f$ is a closed map and hence a homeomorphism. 
One might not prefer this as a proof for your specific problem, though, because one still needs to verify properness of your map $f : (a,b) \to \mathbb{R}$, and my guess is that any argument for properness would be similar to one of the existing answers by others. 
This theorem is an exercise, for example, in Munkres' topology book.