TITLE: Is there a bijective continuous function $f: \Bbb R \to \Bbb R$ whose inverse $f^{-1}$ is not continuous?
QUESTION [8 upvotes]: Looking at the definition of an homeomorphism, this question came to my mind.

REPLY [8 votes]: No. Hint: A bijective continuous function on $\mathbb{R}$ must be monotonic.

REPLY [3 votes]: The answer is no, as can be seen from the Invariance of domain. The answer remains no even when $\mathbb R$ is replaced with $\mathbb R^n$.