Dirichlet function

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It is easy to see that the domain of this function is

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The value range is

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　　The Dirichlet function is a piecewise function . Although we cannot give an intuitive image representation of it, its function image exists objectively. And it can be further proved:

　　(1) The Dirichlet function is an even function , and its image is symmetrical about axis ;

　　(2) The Dirichlet function is a periodic function , and any positive or negative rational number is its period, and there is no minimum positive period.

　　In fact, the Dirichlet function has more mathematical analysis properties. Please refer to the comments on this page. The relevant properties will be given in the future.