上、下半连续性关系及其在经济上的应用

The Relationship Between Upper and Lower Semi-Continuity and Their Application in Economy

上、下半连续性在数学中的重要性不言而喻，在微观经济分析中也有着广泛应用，特别是静态优化问题。分别在单值映射、集值映射中探讨了上半连续性和下半连续性的关系。先证明了单值映射上、下半连续性等价的结论（定理1），并利用引理1对常见函数的上、下半连续性进行了探讨以进一步说明定理1；然后通过举反例进行论证，得出了集值映射中上、下半连续性不等价的结论（定理2）；最后例举了上、下半连续性在数理经济上的应用，具有创新价值。通过对数理经济学中参数约束最优化问题的最大值定理（引理2）条件和结论所做的两点注记。并附以具体实例予以解释，说明了单、集值映射中上、下半连续性的关系，以及在数理经济上的重要应用。

The upper and lower semi-continuity is of self-evident importance to mathematics, also has a wide range of applica- tions in the micro-economic analysis, especially in the static optimization. The relationship between upper and lower semi-continuity is discussed on single-valued mapping and set-valued mapping, respectively. The conclusion that the upper semi-continuity is equal to the lower one is proved on single-valued mapping（ Theorem 1 ）, then Lemma 1 is used to discuss the upper and lower semi-continuity of common function for further elucidating Theorem 1, and the conclusion that the upper semi-continuity on set-valued mapping is non-equal to the lower one there is obtained by illumination with some counter-examples （ Theorem 2 ）. Finally, some examples are used to explain the application of both semi-continuities in the mathematical economy, which is the innovation in the research. Two notes to the condition and conclusion of maximum-value theorem （ Lemma 2） of parameter constrained optimization problem in the mathematical economy and the illustration with specific examples are used to explain the relationship between the upper and lower semi-continuity on both single-valued and set-valued mapping and its important application in mathematical economy.