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在中等数学及数学分析的內容中,我們知道,定义在整个数軸上的以e为底的指数函数e~x具有下面的两个特性: Ⅰ) 对任意的实数x_1,x_2有e~x1·e~x2=e~x1~(+x)2; Ⅱ) (?)en-1/x=1。 現在我們提出一个反面問題,即一个定义在整个数軸上,且又滿足性貭Ⅰ),Ⅱ)的函数是否就必为指数函数e~x呢?答案是肯定的,証明如下: 設E(x)是定义在整个数軸上,滿足关系Ⅰ)E(x_1)。E(x_2)=E(x_1+x_2)及Ⅱ) (?)E(x)-1/x=1的一个函数,則有定理1.E(x)在整个数軸上可导,且E′(x)=E(x)成立。 証.设x是数軸上任意給定的一点,由Ⅰ)有E(x
In the content of medium mathematics and mathematical analysis, we know that the e-base exponential function e~x defined on the whole number axis has the following two characteristics: I) For any real number x_1, x_2 has e~x1 · e~x2=e~x1~(+x)2; II) (?)en-1/x=1. Now we propose a negative question: Is a function defined on the whole number axis and satisfies 貭I), II) whether it must be an exponential function e~x? The answer is affirmative, and the proof is as follows: Let E ( x) is defined on the entire axis and satisfies the relationship I)E(x_1). A function of E(x_2)=E(x_1+x_2) and II)(?)E(x)-1/x=1, then there is theorem 1. E(x) is guidable over the whole number axis, and E ’(x)=E(x) holds. Proof. Let x be any given point on the number axis, by I) with E(x