an example of a continuous bijection which is not a homeomorphism

Question

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is necessarily a homeomorphism)

Any help is appreciated, Thanks !

Answer 1

$X=\mathbb{R}$ with discrete topology and $Y=\mathbb{R}$ with the usual topology, and $f=id$. $f$ is continuous bijective and $f^{-1}=id$ is not continuous.

Answer 2

Using the usual topologies $f:[0,2\pi)\to S^1$ given by $f(x)=e^{ix}$, half open interval to a circle. Continuous, bijective, inverse is discontinuous at $1$, has to break the circle somewhere.