部分翻译
只有在19世纪初期现代分析出现以后?函数的概念才得以扩大。 在扩大的意义上讲函数可定义如下?如果一变量y随着另一个变量x而变换?即x的每一个值都和y的一定值相对应?那么?y就是x的函数。这个定义甚至在今天还适用于许多实际的用途。6
Not specified by this definition is the manner of setting up the correspondence. It may be done by a formula as the 18thcentury mathematics presumed but it can equally well be doneby a tabulation such as a statistical chart, or by some other form of description.
至于如何建立这种对应关系?这个定义并未详细规定。可以如18世纪的数学所假定的那
样?用公式来建立?但同样也可以用统计表那样的表格或用某种其他的描述方式来建立。
A typical example is the room temperature, which obviously isa function of time. But this function admits of no formularepresentation, although it can be recorded in a tabular form or traced but graphically by an automatic device.
典型的例子是室温?这显然是一个时间的函数。但是这个函数不能用公式来代表?但可以用表格的形式来记录或者用一种自动装置以图标形式来追踪
The modern definition of a functionyofxis simply a mapping from a space X to another space Y. a mapping is defined whenevery pointxof X has a definition imagey, a point of Y. the mapping concept is close to intuition, and therefore desirable to serve as a basis of the function concept, Moreover, as the spaceconcept is incorporated in this modern definition, its generalitycontributes much to the generality of the function concept.
现代给x的一个函数y所下的定义只是从一个空间X到另一个空间Y的映射。当X空间的每一个点x有一个确定的像点y?即Y空间的一点?那么?映射就确定了。这个映射概念接近于直观?因此?很可能作为函数概念的一个基础。此外?由于这个现代的定义中体现了空间的概念?所以?它的普遍性对函数概念的普遍性有很大的贡献。