# Does continuous preference imply upper-hemi continuous demand correspondence?

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?

• Debreu's theorem states that continuous preferences have a continuous utility representation, so yeah. Apr 29 at 14:53
• @Giskard Yeah the answer is positive but is it possible to get the result without using utility? Apr 29 at 14:56
• 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
Apr 29 at 16:12
• 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. Apr 29 at 20:45
• @MichaelGreinecker Reading the book right now Apr 30 at 1:16