Existence of utility representation of a rational but discontinuous preference
I think a basic problem is that any utility function defines a preference, and discontinuous utility functions can be used to define discontinuous preferences. Hence there are many discontinuous preferences that can be represented by utility functions.
An example:

Let $U(x,y)$ be a continuous utility function that maps from $\mathbb{R}^2$ to $(0,1)$. This latter may seem arbitrary, but the strictly monotononicaly increasing function $\frac{x}{x+1}$ maps from $\mathbb{R}_{++}$ to $(0,1)$, so it should be fine. Also define a closed set $H \subset \mathbb{R}^2$. Let
$$
\hat{U}(x,y)
=
\left\{
\begin{array}{ll}
U(x,y) &amp; \mbox{if} (x,y) \notin H \\
\\
U(x,y)+1 &amp; \mbox{if} (x,y) \in H.
\end{array}
\right.
$$
Obviously $\hat{U}(x,y)$ is not continuous and neither are the preferences defined by it. (The boundary of $H$ is prefered to anything outside $H$.) But the way these preferences were generated seems fairly general. So a large class of discontinuous preferences exists for which there is a utility representation.

Future question: How 'large' is this class, what measure can be used?

EDIT: As @NicolasPinto points out in his answer it is also necessary to specify that $H$ is such that
$$
\exists x \in H, \exists y \notin H: y \succ x,
$$
so $H$ is not the upper contour set of some point,
otherwise $U(x,y)$ and $\hat{U}(x,y)$ would in fact represent the same continuous preferences.