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关于周期函数,中学课本中已有明确定义,这里不再赘述。而贵刊1994年第9期P.37页刊登的《周期函数》中,是这样定义周期函数的:“对于函数f(x),如果存在一个不为零的常数T,且 (1)对于函数定义域中自变量x的任意值,x+T和x-T都属于函数的定义域; (2)对于函数定义域中的任意x,都有f(x+T)=f(x)或f(x-T)=f(x),则称函数f(x)是以T为周期的周期函数。 笔者认为,这个定义存在两个问题,一是条件(1)是多余的,不符合对一个概念下定义的原则。因为由(2)f(x+T)=f(x)或f(x-T)=f(x)可知x+T或x-T应属于定义域,否则其函数值谈不上相等。二是在(1)中说“x+T和x-T都属于这个函数的定义域”,这又增加了限制条件,从而缩小了概念的外延。实际上x+T和x-T不要求都属于这个函数的定义域,x+T和x-T中有一个属于定义域即可。如,对于函数f(x)=
With regard to the periodic function, there is a clear definition in the middle school textbooks. We will not repeat them here. In “Period Function” published in P.37 of the 9th issue of your magazine, 1994, the periodic function is defined as: “For a function f(x), if there is a constant T that is not zero, and (1) for The arbitrary value of the argument x in the function definition domain, x+T and xT belong to the domain of the function; (2) For any x in the function definition domain, there is f(x+T)=f(x) or f (xT)=f(x), then the function f(x) is said to be a periodic function with period T. The author believes that there are two problems with this definition: First, condition (1) is redundant and does not conform to a concept. The principle defined below, because (2)f(x+T)=f(x) or f(xT)=f(x), we know that x+T or xT should belong to the domain of definition, otherwise the function value is not equal. The second is that in (1) we say that “x+T and xT belong to the domain of this function”, which in turn increases the constraint conditions, thereby reducing the extension of the concept. Actually, x+T and xT do not necessarily belong to this definition. The domain of the function, one of x+T and xT belongs to the definition domain, eg, for the function f(x)=