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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?

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    $\begingroup$ Debreu's theorem states that continuous preferences have a continuous utility representation, so yeah. $\endgroup$
    – Giskard
    Apr 29, 2022 at 14:53
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    $\begingroup$ 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. $\endgroup$
    – Amit
    Apr 29, 2022 at 16:12
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    $\begingroup$ 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. $\endgroup$ Apr 29, 2022 at 20:45
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    $\begingroup$ @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!! $\endgroup$
    – T123
    Sep 29, 2022 at 11:53
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    $\begingroup$ @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. $\endgroup$
    – High GPA
    Dec 11, 2022 at 13:50

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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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