# Debreu's Representation Theorem proof

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?

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

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 $$u : X \to \mathbb R$$.