论文部分内容阅读
设N是一个实数等。定义在N上的函数f(x);若对某一数r(?)0,具有性貭:(1) 对于任一点x∈N,x±r∈N;(2) f(x+r)=f(x)在N上恆成立;那末就称f(x)为集N上的周期函数,r叫做函数f(x)的周期。根据这一定义显然可見: (一) 若r(?)0是f(x)的周期,則-r也是f(x)的周期,事实上,在所給条件下,有f(x-r)=f(x-r++r)=f(x)在N上恆成立。 (二) 若r(?)0是f(x)的周期,则nr(n:任意的自然数)也是f(x)的周期,这是因为在所給条件下,有f(x++nr)=f(x+n=1r+r)=f(x=n-1r)=…==f(x+r)=f(x)在N上恆成立。总括(一),(二)可見,周期函数的一切周期組成了一个关于原点成对称的无穷集合;因此,对周期函数的周期进行研究时,但研究其正的周期就够了。但即使对于定义在整个数軸上的周期函数的所有正周期而言;并不是都有最小的,例如定义在整个数軸上处处不連續的狄里克萊函数
Let N be a real number and so on. The function f(x) defined on N; if for a certain number r(?)0, it has the property 貭: (1) for any point x∈N, x±r∈N; (2) f(x+r )=f(x) is constant on N; then we call f(x) a periodic function on set N and r is called the periodicity of function f(x). According to this definition, it is obvious that: (1) If r(?)0 is the period of f(x), then -r is also the period of f(x). In fact, under the given conditions, there is f(xr)= f(x-r++r)=f(x) holds true on N. (2) If r(?)0 is the period of f(x), then nr(n: any natural number) is also the period of f(x), because under the given conditions, there is f(x++nr) )=f(x+n=1r+r)=f(x=n-1r)=...==f(x+r)=f(x) holds true on N. In summary (1) and (2), it can be seen that all cycles of the periodic function form an infinite set that is symmetrical about the origin; therefore, when the periodicity of the periodic function is studied, it is sufficient to study its positive periodicity. But even for all positive cycles of the periodic function defined over the whole number axis, there is not a minimum, for example, a Dirichlet function defined everywhere in the whole number axis.