怎么证明
Proposition 2.9 (Debreu's Theorem). If a continuous function $u$ represents $\succeq$, then $\succeq$ is continuous. Conversely, if $\succeq$ is continuous, it has a continuous utility representation. ${ }^{5}$

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∻Answer∻ ‖ a ⋮ Debreu's Theorem states that if a continuous function $u$ represents a preference relation $\succeq$, then the preference relation $\succeq$ is continuous. This means that small changes in the consumption bundle lead to small changes in the utility level. ‖‖ b ⋮ Conversely, if the preference relation $\succeq$ is continuous, it implies that there exists a continuous utility function $u$ that can represent it. This is crucial in microeconomic theory as it links preferences and utility in a continuous manner. ‖ ∻Answer∻ ⚹ Debreu's Theorem connects continuity in preferences and utility representation. ⚹ ∻Key Concept∻ ⚹ Debreu's Theorem ⚹ ∻Explanation∻ ⚹ It establishes the relationship between continuous preferences and the existence of a continuous utility function. ⚹◊What conditions must be satisfied for a preference relation to have a continuous utility representation according to Debreu's Theorem?⍭ Generate me a similar question◊

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