Can a continuous preference be represented by a discountinuous function?

Question

I can think of some examples, but what can be an outline of the proof?

Answer 1

Indeed you are correct that a continuous utility function can be represented by a discontinuous function. However, I am not sure what you mean by proof beyond an example; an example is a proof of this fact.

For completeness, here is an example we can take $\succeq\,\subset \mathbb R \times \mathbb R$ to reflect the usual ordering (i.e., $x \succeq y$ iff $x \geq y$). Clearly this is a continuous ordering as its contour sets are intervals. The utility function $$ U(x) = \begin{cases} x &\text{ if } x < 4 \\ x + 1 &\text{ if } x \geq 4 \end{cases} $$ is discontinuous, but represents $\succeq$.

Any such function is necessarily have a continuous representation as well.