近似连续点与Lebesgue点的性质及其应用

Approximate continuous point and Lebesgue point with its properties and application

利用实分析、测度论中的一些处理技巧,首先证明得知不定积分在定义域中的任意Lebesgue点处必可导,同时结合相关文献指出相应的逆命题不成立.其次分析近似连续点、Lebesgue点、连续点、间断点、孤立点之间的相互关系.最后通过对Dirichlet函数与Riemann函数的研究分析,得出两类函数的近似连续点与Lebesgue点的分布情况,并作为反例否定了一些结论的逆命题,进一步拓展了两类函数的一些性质.

By using some processing techniques in real analysis and measure theory,first,it is proved that the indefinite integral must be guided at any Lebesgue point in the domain,and according to related literature,the corresponding some inverse propositions are not established. Then,research the relations among approximately continuous points,Lebesgue points,continuous points,inter breakpoints,and isolated points. Finally,through research Dirichlet function and Riemann function,the distributions of approximate continuous points and Lebesgue points of two classes of functions are obtained,The negative propositions of some conclusions are negated as negative examples,some properties of the two classes of functions are further extended.