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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 at 14:53
  • $\begingroup$ @Giskard Yeah the answer is positive but is it possible to get the result without using utility? $\endgroup$
    – High GPA
    Apr 29 at 14:56
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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 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 at 20:45
  • $\begingroup$ @MichaelGreinecker Reading the book right now $\endgroup$
    – High GPA
    Apr 30 at 1:16

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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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