2
$\begingroup$

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

$\endgroup$
  • 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 Oct 16 '18 at 14:55
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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