1. If the function f(x)(x∈D) has two symmetry axes in the domain, x=a and x=b, then the function f(x) is a periodic function, and the period T=2|b-a| (not necessarily the minimum positive period).
2. If the function f(x)(x∈D) has two symmetrical centers A(a, 0) and B(b, 0) in the definition domain, then the function f(x) is a periodic function, and the period T=2|b-a| (not necessarily the minimum positive period).
3. If the function f(x)(x∈D) has an axis of symmetry x=a and a center of symmetry B(b, 0)(a≠b), then the function f(x) is a periodic function with a period T=4|b-a| (not necessarily the minimum positive period).
Let the extended data set f(x) be a function defined on the number set m. If there is a non-zero constant t whose property is: f(x+T)=f(x), it is said that f(x) is a periodic function on the number set m, and the constant t is called a period of f(x). If there is a minimum in all positive periods, it is called the minimum positive period of the function f(x).
The properties of the periodic function are as follows:
(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*.
Baidu encyclopedia-function periodicity