<p>To get you started:</p>
<p>We are <em>told</em> that this utility function represents a week preference relation.</p>
<p>Then (<strong>transitivity</strong>)</p>
<p>For <span class="math-container">$x_1, x_2, x_3 \in X$</span> suppose we get
<span class="math-container">$$u(x_1)\geq u(x_2)\geq u(x_3).$$</span>
These are real numbers, and the chain weak inequality implies also
<span class="math-container">$$u(x_1) \geq u(x_3)$$</span></p>
<p>Since we have been told that this <span class="math-container">$u$</span> represents a preference relation, then it must be the case that
<span class="math-container">$$x_1 \succsim x_2,\;\; x_2 \succsim x_3,\;\; x_1 \succsim x_3. $$</span>
This proves that this preference relation that is represented by this utility function is transitive.</p>
<p><strong>Completeness</strong>
When you look at <em>any</em> two real numbers, is there a case where you are unable to say whether the one is higher than, lower than, or equal to, the other?</p>
<p><strong>Continuity</strong>
You should follow the advice given in a comment to your post, which essentially tells you to put side by side the definitions of continuity of a real-valued function and that of preferences.</p>

CC BY-SA 4.0
2022-08-20T20:17:49.620Z
2022-08-20T20:17:49.620Z
Alecos Papadopoulos
<p>The reverse of Debreu Theorem, have no idea how to start the question.</p>
<p><a href="https://i.sstatic.net/86en2.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/86en2.png" alt="original question" /></a></p>

CC BY-SA 4.0
2022-08-20T12:28:23.533Z
Can you please define the problem, with some notation.
BrsG
CC BY-SA 4.0
2022-08-20T13:39:57.263Z
hi, I just add the original question. thx
chris
CC BY-SA 4.0
2022-08-20T14:28:44.703Z
Hi! Can you please include the definitions of these concepts (cont. preferences, cont. U func.) in your question?
Giskard
<microeconomics>
How to prove that preference relation is continuous if the utility function represent the preference relation is continuous
chris
# How to prove that preference relation is continuous if the utility function represent the preference relation is continuous

(Asked by: username_0 on 2022-08-20)

The reverse of Debreu Theorem, have no idea how to start the question.

[![original question](https://i.sstatic.net/86en2.png)](https://i.sstatic.net/86en2.png)

**Tags:** microeconomics

> **Comments:**
> **username_1**: Can you please define the problem, with some notation.  
> **username_0**: hi, I just add the original question. thx  
> **username_2**: Hi! Can you please include the definitions of these concepts (cont. preferences, cont. U func.) in your question?  

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## 1 answers

### Answer by username_3 on 2022-08-20. Score: 3 | Accepted answer

To get you started:

We are *told* that this utility function represents a week preference relation.

Then (**transitivity**)

For $x\_1, x\_2, x\_3 \in X$ suppose we get
$$u(x\_1)\geq u(x\_2)\geq u(x\_3).$$
These are real numbers, and the chain weak inequality implies also
$$u(x\_1) \geq u(x\_3)$$

Since we have been told that this $u$ represents a preference relation, then it must be the case that
$$x\_1 \succsim x\_2,\;\; x\_2 \succsim x\_3,\;\; x\_1 \succsim x\_3. $$
This proves that this preference relation that is represented by this utility function is transitive.

**Completeness**
When you look at *any* two real numbers, is there a case where you are unable to say whether the one is higher than, lower than, or equal to, the other?

**Continuity**
You should follow the advice given in a comment to your post, which essentially tells you to put side by side the definitions of continuity of a real-valued function and that of preferences.


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