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Pringsheim Convergence and the Dirichlet Function

机译:普林斯海姆收敛与狄利克雷函数

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摘要

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and ?exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.
机译:双序列具有一些意外的特性,这些特性源于通勤极限操作的可能性。例如,可以定义为使得对所有x的迭代限制和都存在且相等,而普林斯海姆限制不存在。该序列是一个经典示例,用于显示连续函数的双重序列的迭代极限可能存在,但会导致无处不在的极限。我们探索了在普林斯海姆意义上该序列的极限是否等于迭代结果,并得出了余弦作为副产物的有趣性质。

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