<p>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 <strong>is</strong> a proof of this fact. </p>

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

<p>Any such function is necessarily have a continuous representation as well. </p>

CC BY-SA 4.0
2022-10-04T02:13:52.980Z
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](https://economics.stackexchange.com/q/52947/32979).
Dave
CC BY-SA 4.0
2018-10-16T14:36:07.090Z
2018-10-16T14:36:07.090Z
201p
<p>I can think of some examples, but what can be an outline of the proof?</p>

CC BY-SA 4.0
2018-10-16T14:55:16.060Z
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
<utility><preferences>
Can a continuous preference be represented by a discountinuous function?
plastico
# Can a continuous preference be represented by a discountinuous function?

(Asked by: username_0 on 2018-10-12)

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

**Tags:** utility, preferences
> **Comments:**
> **username_1**: A continuous preference relation can always be represented by a continuous function. However,  It can also be represented bu other (nom-continuous) functions.  

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## 1 answers

### Answer by username_2 on 2018-10-16. Score: 6 | Accepted 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.

> **Comments:**
> **username_3**: 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](https://economics.stackexchange.com/q/52947/32979).  

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