TITLE: continuous invertible map discontinuous inverse
QUESTION [7 upvotes]: Is there an example of a continuous invertible map $f:X\to Y$ between topological spaces $(X,T_X)$ and $(Y,T_Y)$ such that $f$ is continuous, but its inverse $f^{-1}$ is not continuous?

REPLY [2 votes]: Using discrete metric is the easiest examplethat you can think of a continuous biyection that is not a homeomorphism but is some artificial. Hardly expect that a case like this occurs during the solution of a problem of analysis or geometry.
An example more natural is the following. Let $M=[-1,0]\cup(1,\infty)$ and $N=[0,\infty)$ and let $f:M\to N$ defined by $f(x)=x^2$. You can check easly that $f$ is continuous bijection (the graph has 2 separated parts but you don't surprise with this fact because domain has two disjoint parts). But $f^{-1}$ is discontinuous at $y=1$ and domain of inverse has only one part, but graph has two: