TITLE: Is the inverse of a continuous bijective function also continuous?
QUESTION [9 upvotes]: Is the inverse of a continuous bijective function also continuous?
How to prove it?

REPLY [5 votes]: Take any set $S$. Let $X$ be $S$ with the discrete topology and $Y$ be $S$ with the coarse topology. Note that the identity $i:X\to Y$ is continuous, but its inverse, the identity $i:Y\to X$, is not.