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In this question, it is shown that a continuous preference relation can have a discontinuous utility function.

Is it true in general that every continuous preference relation must have a discontinuous utility function?

I know that a continuous preference relation must have a continuous utility representation, and I am trying to compose such a continuous function with an increasing but discontinuous function (which will be a utility function for the preference relation), but since the composition of a continuous and discontinuous function can be continuous, it is not trivial to show this composition to be discontinuous.

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That is not true. Consider the continuous preference relation on $\mathbb{R}^n_+$ represented by utility function $u(x)= 0$. This does not have a discontinuous utility representation. In fact, this is the only continuous preference relation on $\mathbb{R}^n_+$ that does not have a discontinuous utility representation.

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  • $\begingroup$ Would this mean that $\succsim=\mathbb R_+^n\times\mathbb R_+^n$ so that there is perfect indifference: $x\sim y$ for all $x,y\in\mathbb R_+^n?$ $\endgroup$
    – Dave
    Oct 4, 2022 at 3:22
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    $\begingroup$ Yes that's right. $\endgroup$
    – Amit
    Oct 4, 2022 at 3:23

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