<p>I think I find the solution: this short paper includes a generalized version of Berge's Theorem <a href="https://www.jstor.org/stable/2526431?seq=1" rel="nofollow noreferrer">https://www.jstor.org/stable/2526431?seq=1</a></p>
<p>The binary relation is used instead of utility. The binary relation does not need to be transitive or irreflective. Some mild assumption of relaxed completeness is still needed.</p>

CC BY-SA 4.0
2022-05-02T05:59:32.480Z
2022-05-02T09:02:02.637Z
2022-05-02T09:02:02.637Z
High GPA
<p>Let alternative <span class="math-container">$x,y,z\in R^N$</span>.</p>
<p><span class="math-container">$\succsim$</span> is convex, rational, monotonic, and continuous.</p>
<p>Let <span class="math-container">$B=[y,z]$</span> be a budget segment.</p>
<p>Let demand correspondence be <span class="math-container">$D[y,z]=\{x\in B||x\succsim B\}$</span></p>
<p><span class="math-container">$D[y,z]$</span> is a set. <span class="math-container">$D$</span> is continuous if a small change in <span class="math-container">$y$</span> causes a small change in all <span class="math-container">$x\in D[y,z]$</span>.</p>
<p>Suppose that demand correspondence is defined with utility, then the demand is upper-hemi continuous by Berge's theorem.</p>
<p>However, without using utility function, can we have a similar result?</p>

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2022-04-29T14:53:22.837Z
Debreu's theorem states that continuous preferences have a continuous utility representation, so yeah.
Giskard
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2022-04-29T14:56:23.543Z
@Giskard Yeah the answer is positive but is it possible to get the result without using utility?
High GPA
CC BY-SA 4.0
2022-04-29T16:12:58.473Z
Yes, you just need to rewrite the proof of the theorem using the preferences directly. This is because the demand correspondence will not change whether you work on the preference or its utility representation. So the result will not change.
Amit
CC BY-SA 4.0
2022-04-29T20:45:23.783Z
Chapter 12 of Kim Border's "Fixed point theorems with applications to economics and game theory" has some generalizations of Berge's maximum theorem that work for more general preferences than those that have utility representations.
Michael Greinecker
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2022-04-30T01:16:42.770Z
@MichaelGreinecker Reading the book right now
High GPA
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2022-09-29T11:53:01.843Z
@HighGPA I am fascinated by your questions as i'm also interested in the field of preferences, utility functions and choice theory. As an avid reader of your questions and remarks: What do you read and where did you get all of this from? Any tips for me where to start reading? Thanks!!
T123
CC BY-SA 4.0
2022-12-11T13:50:33.993Z
@T123 Hi, thanks for your comment. I am not very creative so my thinking usually stems from modification of popular textbook theorems. For example, I will think about what if the continuity is dropped in the vNM utility theorem, and what will happen if some assumptions are dropped in the Berge's theorem.
High GPA
<microeconomics><preferences><theory><demand>
Does continuous preference imply upper-hemi continuous demand correspondence?
High GPA
# Does continuous preference imply upper-hemi continuous demand correspondence?

(Asked by: username_0 on 2022-04-29)

Let alternative $x,y,z\in R^N$.

$\succsim$ is convex, rational, monotonic, and continuous.

Let $B=[y,z]$ be a budget segment.

Let demand correspondence be $D[y,z]=\{x\in B||x\succsim B\}$

$D[y,z]$ is a set. $D$ is continuous if a small change in $y$ causes a small change in all $x\in D[y,z]$.

Suppose that demand correspondence is defined with utility, then the demand is upper-hemi continuous by Berge's theorem.

However, without using utility function, can we have a similar result?

**Tags:** microeconomics, preferences, theory, demand
> **Comments:**
> **username_1**: Debreu's theorem states that continuous preferences have a continuous utility representation, so yeah.  
> **username_0**: @Giskard Yeah the answer is positive but is it possible to get the result without using utility?  
> **username_2**: Yes, you just need to rewrite the proof of the theorem using the preferences directly. This is because the demand correspondence will not change whether you work on the preference or its utility representation. So the result will not change.  
> **username_3**: Chapter 12 of Kim Border's "Fixed point theorems with applications to economics and game theory" has some generalizations of Berge's maximum theorem that work for more general preferences than those that have utility representations.  
> **username_0**: @MichaelGreinecker Reading the book right now  
> **username_4**: @HighGPA I am fascinated by your questions as i'm also interested in the field of preferences, utility functions and choice theory. As an avid reader of your questions and remarks: What do you read and where did you get all of this from? Any tips for me where to start reading? Thanks!!  
> **username_0**: @T123 Hi, thanks for your comment. I am not very creative so my thinking usually stems from modification of popular textbook theorems. For example, I will think about what if the continuity is dropped in the vNM utility theorem, and what will happen if some assumptions are dropped in the Berge's theorem.  

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## 1 answers

### Answer by username_0 on 2022-05-02. Score: 2

I think I find the solution: this short paper includes a generalized version of Berge's Theorem <https://www.jstor.org/stable/2526431?seq=1>

The binary relation is used instead of utility. The binary relation does not need to be transitive or irreflective. Some mild assumption of relaxed completeness is still needed.


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