# Can a continuous preference be represented by a discountinuous function?

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

• A continuous preference relation can always be represented by a continuous function. However, It can also be represented bu other (nom-continuous) functions. – BB King Oct 16 '18 at 14:55

## 1 Answer

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.