<p>At the heart of Debreu's representation theorem is his so-called &quot;gap theorem&quot;:</p>
<p>Let <span class="math-container">$S\subseteq[0,1]$</span>. A <em>gap</em> is a maximal nontrivial interval disjoint from <span class="math-container">$S$</span> with an upper and lower bound in <span class="math-container">$S$</span>. Debreu's gap theorem says that there is a strictly increasing function <span class="math-container">$f:S\to\mathbb{R}$</span> such that all gaps in the image <span class="math-container">$f(S)$</span> are open intervals. The intuition of Debreu was that if a gap is a half-open interval, then one can slide the endpoints together to remove the gap. His initial proof attempt based on this idea in [Debreu, Gerard. &quot;Representation of a preference ordering by a numerical function.&quot; <em>Decision processes</em> 3 (1954): 159-165.] turned out to be wrong as Debreu himself observed in [Debreu, Gerard. &quot;Continuity properties of Paretian utility.&quot; <em>International Economic Review</em> 5.3 (1964): 285-293.], where he also supplied a very lengthy correct proof. There have been many proofs of the gap theorem since, starting with a slick but nonelementary proof based on measure theory in [Bowen, Robert. &quot;A new proof of a theorem in utility theory.&quot; <em>International Economic Review</em> 9.3 (1968): 374-374.]</p>
<p>Now, why is the gap theorem useful? Take any utility function <span class="math-container">$v:X\to [0,1]$</span> that represents continuous preferences. Let <span class="math-container">$u:X\to\mathbb{R}$</span> be the composition <span class="math-container">$f\circ v$</span> with <span class="math-container">$f$</span> the kind of function guaranteed to exist by the gap lemma. It turns out that <span class="math-container">$u$</span> is then continuous. Clearly, it also represents the preferences. Indeed, it suffices to show that for each <span class="math-container">$r\in\mathbb{R}$</span>, the preimages <span class="math-container">$u^{-1}\big ((-\infty, r]\big)$</span> and <span class="math-container">$u^{-1}\big([r, \infty)\big)$</span> are closed. Essentially, one uses the gap theorem to show these intervals are order-closed. Since preferences are continuous, order-closed sets are actually closed.</p>
<p>It is worth pointing out that this subject is very technical; there is no easy proof for Debreu's theorem in full generality.</p>

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2022-07-31T17:02:49.270Z
2022-07-31T17:02:49.270Z
Michael Greinecker
<p>Suppose that <span class="math-container">$X \subset \mathbb{R}^n$</span>. A preference relation <span class="math-container">$\preceq$</span> is reflexive, complete, transitive and continuous if and only if there exists a utility function <span class="math-container">$u:X \rightarrow \mathbb{R}$</span> that represents it.<br />
How we prove this kind of proof, I saw proof of Debreu Theorem but I don't understand every step and why it's true intuitively.</p>

CC BY-SA 4.0
2022-07-29T09:51:48.757Z
Result is : A preference relation $\preceq$ is reflexive, complete, transitive and continuous if and only if there exists a continuous utility function $u:X \rightarrow \mathbb{R}$ that represents it. Examples posted here (both in the question and the answer) are examples of utility functions that represents discontinuous preference: https://economics.stackexchange.com/q/51772/11824
Amit
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2022-07-29T10:10:43.880Z
@Amit Can you explain briefly why both directions (if and only if) are true, I didn't understand that much even from the examples.
Djedjou Emery
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2022-07-29T11:30:03.543Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Giskard
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2022-07-29T11:53:31.250Z
@Giskard I don't understand why it's true if the preference relation is continuous, and why it's true that it means that exists a continuous utility function.
Djedjou Emery
CC BY-SA 4.0
2022-07-29T11:55:28.990Z
@Giskard More simple that debreu theorem means: " Every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function." My question is why it's true
Djedjou Emery
CC BY-SA 4.0
2022-07-29T13:12:27.453Z
Yes, I got that part, but did you read the proof somewhere? If yes, what part did you not understand? If not, are you asking for a reference of where you can read the proof?
Giskard
CC BY-SA 4.0
2022-07-29T16:13:40.370Z
@Giskard I watch here: https://sites.pitt.edu/~luca/ECON2100/2018Class/lecture_03.pdf from page 11. (Debreu Theorem Proof) (Is one direction trivial? If yes, I don't see it, if you have explanation will be great or another site with a simply proof from this.
Djedjou Emery
CC BY-SA 4.0
2022-07-29T16:18:29.237Z
@DjedjouEmery So, one direction is proven here; do you understand that one? The other direction is not proven at all. Have you tried reading that part of the proof in Debreu's original 1954 paper?
Giskard
CC BY-SA 4.0
2022-07-29T16:19:11.990Z
@Giskard No, do you have maybe a link?
Djedjou Emery
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2022-07-29T16:19:37.130Z
@Giskard And I don't really understand unfortunately...
Djedjou Emery
CC BY-SA 4.0
2022-07-31T17:01:42.503Z
Your question asks for the equivalence of these properties with the existence of any utility function. But you only get continuous preferences when a continuous utility representation exists.
Michael Greinecker
<utility><preferences>
A preference relation is continuous if and if there exists a utility function that represents it
Djedjou Emery
# A preference relation is continuous if and if there exists a utility function that represents it

(Asked by: username_0 on 2022-07-29)

Suppose that $X \subset \mathbb{R}^n$. A preference relation $\preceq$ is reflexive, complete, transitive and continuous if and only if there exists a utility function $u:X \rightarrow \mathbb{R}$ that represents it.  
How we prove this kind of proof, I saw proof of Debreu Theorem but I don't understand every step and why it's true intuitively.

**Tags:** utility, preferences

> **Comments:**
> **username_1**: Result is : A preference relation $\preceq$ is reflexive, complete, transitive and continuous if and only if there exists a continuous utility function $u:X \rightarrow \mathbb{R}$ that represents it. Examples posted here (both in the question and the answer) are examples of utility functions that represents discontinuous preference: https://economics.stackexchange.com/q/51772/11824  
> **username_0**: @username_1 Can you explain briefly why both directions (if and only if) are true, I didn't understand that much even from the examples.  
> **username_2**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  
> **username_0**: @username_2 I don't understand why it's true if the preference relation is continuous, and why it's true that it means that exists a continuous utility function.  
> **username_0**: @username_2 More simple that debreu theorem means: " Every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function." My question is why it's true  
> **username_2**: Yes, I got that part, but did you read the proof somewhere? If yes, what part did you not understand? If not, are you asking for a reference of where you can read the proof?  
> **username_0**: @username_2 I watch here: https://sites.pitt.edu/~luca/ECON2100/2018Class/lecture_03.pdf from page 11. (Debreu Theorem Proof) (Is one direction trivial? If yes, I don't see it, if you have explanation will be great or another site with a simply proof from this.  
> **username_2**: @DjedjouEmery So, one direction is proven here; do you understand that one? The other direction is not proven at all. Have you tried reading that part of the proof in Debreu's original 1954 paper?  
> **username_0**: @username_2 No, do you have maybe a link?  
> **username_0**: @username_2 And I don't really understand unfortunately...  
> **username_3**: Your question asks for the equivalence of these properties with the existence of any utility function. But you only get continuous preferences when a continuous utility representation exists.  

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## 1 answers

### Answer by username_3 on 2022-07-31. Score: 3 | Accepted answer

At the heart of Debreu's representation theorem is his so-called "gap theorem":

Let $S\subseteq[0,1]$. A *gap* is a maximal nontrivial interval disjoint from $S$ with an upper and lower bound in $S$. Debreu's gap theorem says that there is a strictly increasing function $f:S\to\mathbb{R}$ such that all gaps in the image $f(S)$ are open intervals. The intuition of Debreu was that if a gap is a half-open interval, then one can slide the endpoints together to remove the gap. His initial proof attempt based on this idea in [Debreu, Gerard. "Representation of a preference ordering by a numerical function." *Decision processes* 3 (1954): 159-165.] turned out to be wrong as Debreu himself observed in [Debreu, Gerard. "Continuity properties of Paretian utility." *International Economic Review* 5.3 (1964): 285-293.], where he also supplied a very lengthy correct proof. There have been many proofs of the gap theorem since, starting with a slick but nonelementary proof based on measure theory in [Bowen, Robert. "A new proof of a theorem in utility theory." *International Economic Review* 9.3 (1968): 374-374.]

Now, why is the gap theorem useful? Take any utility function $v:X\to [0,1]$ that represents continuous preferences. Let $u:X\to\mathbb{R}$ be the composition $f\circ v$ with $f$ the kind of function guaranteed to exist by the gap lemma. It turns out that $u$ is then continuous. Clearly, it also represents the preferences. Indeed, it suffices to show that for each $r\in\mathbb{R}$, the preimages $u^{-1}\big ((-\infty, r]\big)$ and $u^{-1}\big([r, \infty)\big)$ are closed. Essentially, one uses the gap theorem to show these intervals are order-closed. Since preferences are continuous, order-closed sets are actually closed.

It is worth pointing out that this subject is very technical; there is no easy proof for Debreu's theorem in full generality.


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