For the function y=f(x), if there is a non-zero constant t, so that f(x+T)=f(x) holds when x takes every value in the domain, then the function y=f(x) is called a periodic function, and the non-zero constant t is called the period of this function.

Properties of periodic functions:

(1) If T(≠0) is the period of f(X), then -T is also the period of f(X).

(2) If T(≠0) is the period of f(X), then nT(n is an arbitrary non-zero integer) is also the period of f(X).

(3) If T 1 and T2 are both periods of f(X), then T 1 T2 is also a period of f(X).

(4) If f(X) has a minimum positive period T*, then any positive period t of f(X) must be a positive integer multiple of T*.

(5)T* is the minimum positive period of f(X), and T 1 and T2 are two periods of f(X) respectively, then (q is a rational number set).

(6) If T 1 and T2 are two periods of f(X) and are irrational numbers, then f(X) does not have a minimum positive period.

(7) The domain m of the periodic function f(X) must be a set with unbounded sides.