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

CC BY-SA 4.0
2022-10-04T03:22:07.380Z
Would this mean that $\succsim=\mathbb R_+^n\times\mathbb R_+^n$ so that there is perfect indifference: $x\sim y$ for all $x,y\in\mathbb R_+^n?$
Dave
CC BY-SA 4.0
2022-10-04T03:23:15.627Z
Yes that's right.
Amit
CC BY-SA 4.0
2022-10-04T03:11:35.263Z
2022-10-04T03:11:35.263Z
Amit
<p>In <a href="https://economics.stackexchange.com/q/24983/32979">this question</a>, it is shown that a continuous preference relation can have a discontinuous utility function.</p>
<p>Is it true in general that every continuous preference relation must have a discontinuous utility function?</p>
<p>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 <a href="https://math.stackexchange.com/q/1655958/672549">the composition of a continuous and discontinuous function can be continuous</a>, it is not trivial to show this composition to be discontinuous.</p>

<utility><preferences>
Can every continuous preference relation be represented by a discontinuous utility function?
Dave
# Can every continuous preference relation be represented by a discontinuous utility function?

(Asked by: username_0 on 2022-10-04)

In [this question](https://economics.stackexchange.com/q/24983/32979), 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](https://math.stackexchange.com/q/1655958/672549), it is not trivial to show this composition to be discontinuous.

**Tags:** utility, preferences


---

## 1 answers

### Answer by username_1 on 2022-10-04. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: Would this mean that $\succsim=\mathbb R_+^n\times\mathbb R_+^n$ so that there is perfect indifference: $x\sim y$ for all $x,y\in\mathbb R_+^n?$  
> **username_1**: Yes that's right.  

---

