$\succsim$ is a weak order over $\mathbb R^L$.

For a closed budget set $B\subset\mathbb R^L$, define demand correspondence: $$D(B)=\{x\in B|x\succsim y\forall y\in B\}$$.

We know that $D$ is always none empty and both upper and lower hemi-continuous, can we conclude that $\succsim$ is continuous? Can we conclude that $D$ is rationalizable by a utility function?

My approach: For the first question, I guess we can first show that $D$ is rationalizable by a utility function $u$. Secondly, since $D$ is continuous, $u$ must be continuous, and $\succsim$ is continuous. But I doubt this approach will work, because the second step seems not work.

For the second part, if $\succsim$ is continuous then it is represented by a continuous utility function. Even if $\succsim$ is not continous, it must be upper-semi continuous and we can still find a upper semi-continuous utility rationalization for $D$ by Rader theorem.

We could consider a simpler version of the problem: Let the demand function be continuous, is the preference relation also continuous? (This is proved by Mas-Colell in 1978, I think)


I think that you should proceed by contradiction assume D is continuous, but $\succsim$ is not, then for a bundle either the more preferred than or the less preferred than sets are not closed. Choose the problematic set and choose a set B appropriately to get a contradiction of the existence of a maximum (remember that not closed sets might not admit a maximum), this will give you the desired contradiction. Once preferences are continuous, remains to check completeness of preferences to use the classic theorem that if preferences are complete, transitive and continuous, then a utility function representation exists.

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