In the teaching examples of piecewise functions in this section, we have introduced Dirichlet functions. Here, we want to prove that the Dirichlet functionIn the teaching examples of

,

is a periodic function .

　　Proof: First prove that any positive rational number is its period. Suppose is a positive rational number and is any real number. Then when is a rational number, is also a rational number; when is an irrational number , is also an irrational number, that is, we have

,

Therefore, the positive rational number is the period of the Dirichlet function .

　　Note that if is a positive rational number, then is a negative rational number. It can be proved exactly similar to the above that any negative rational number is also its period. Therefore, any non-zero rational number is the period of the Dirichlet function. But since there is no minimum positive rational number, does not have a minimum positive period .