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In microeconomy this theorem states that : given a consumption set $X\subseteq\mathbb{R}^n$, if the preference relation $\succcurlyeq$ is complete, transitive and continuous there exist a utility function $u$ and it is continuous. I've no problem at all in the first part, but at the end, to prove the continuity of $u$ it is said that it suffices to show that $u^{-1}(a,b)$ is open for all $a$,$b \in\mathbb{R}$. Well i don't get this part, is there somebody who can help me?

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  • $\begingroup$ Where did you find this proof? $\endgroup$
    – Bob Jansen
    Feb 8, 2018 at 12:08
  • $\begingroup$ My teacher told this during lesson $\endgroup$
    – ab94
    Feb 8, 2018 at 14:33
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    $\begingroup$ This is Proposition 3.C.1 in Mas-Colell et al. "Microeconomic Theory", which also contains the corresponding proof. $\endgroup$ Feb 8, 2018 at 14:57
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    $\begingroup$ I'm voting to close this question as off-topic because it is about economics and not quant finance. $\endgroup$ Feb 9, 2018 at 11:13
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    $\begingroup$ Personally, I don't mind questions about mathematical economics. Our user base seems to be well-suited for this type of question. $\endgroup$
    – Bob Jansen
    Feb 18, 2018 at 14:24

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A function $f: X \to Y$ is continuous if for every open set $V$ in $Y$, the preimage $f^{-1}(V)$ is open in $X$.

Any open subset of the reals, which is not the empty set, is an open interval or the union of open intervals. Note that $f^{-1}(A\cup B) = f^{-1}(A) \cup f^{-1}(B)$. Since the union of open sets is open, it suffices to only consider the preimage of an open interval to prove continuity of $u : X \to \mathbb R$.

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  • $\begingroup$ note that an open interval itself is also a union of open intervals :) $\endgroup$ Mar 1, 2022 at 18:34

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