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

  • 1
    $\begingroup$ A continuous preference relation can always be represented by a continuous function. However, It can also be represented bu other (nom-continuous) functions. $\endgroup$
    – BB King
    Commented Oct 16, 2018 at 14:55

1 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.

  • $\begingroup$ I would think that a proof would be showing that, in general, any continuous preference can be represented by a discontinuous utility function. I’ve now posted this as a separate question. $\endgroup$
    – Dave
    Commented Oct 4, 2022 at 2:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.