TITLE: Functions which are Continuous, but not Bicontinuous
QUESTION [62 upvotes]: What are some examples of functions which are continuous, but whose inverse is not continuous?
nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.

REPLY [8 votes]: 1) Take any topological space,
2) Obtain another space by refining its topology,
3) ...
4) PROFIT!
In fact, consider the forgetful functor $F: \mathbf{Top} \to \mathbf{Set}$. For any set $S$ the continuous functions of the form $f: X \to Y$ such that $FX = FY = S$ and $Ff = 1_S$ form a linear order on the set of all topologies on $S$, and this order is in fact inverse to the usual one (formed by set inclusion of topologies).
For example, extending the answer by Marco, consider a simple curve $\gamma: I \to M$ on some manifold with the finite number of self-intersections. For each intersection, remove all corresponding points from $I$ except for one. Voila :) UPD: actually, you can remove all corresponding points, period!

REPLY [6 votes]: Take a "8"-shaped plane curve $\mathcal C \subset \mathbb R^2$ endowed with the subspace topology. 
Let $\phi: \mathbb R \to \mathcal C$ be a continuous injective parametrization of $\mathcal C$. The inverse function $\phi^{-1}: \mathcal   C\to\mathbb R$ cannot be continuous because $\phi^{-1}((a,+\infty))$ is not an open set for some $a$.