TITLE: homeomorphism non-example
QUESTION [10 upvotes]: A homeomorphism is a continuous function between topological spaces that has a continuous inverse function.
Can someone provide examples of a continuous function between topological spaces that does not have a continuous inverse function? 
It would help me grasp the intuition for what a homeomorphism is.
Feel free to use different topological spaces for your examples (but at least one with $\mathbb{R}$ would be nice)!

REPLY [2 votes]: $$
(\theta\in [0,1))\mapsto((\cos\theta,\sin\theta)\in\text{circle}).
$$
This is continuous.
But as you move along the circle, $\theta$ has a jump discontinuity where it jumps from $1$ to $0$ or from $0$ to $1$ (depending on whether you're going clockwise or counterclockwise).