无穷小微积分的基本定理

    五年前，无穷小放飞互联网期间，我们把无穷小微积分学的基本定理放在互联网上，让菲氏微积分守旧派看看。

Suppose f is continuous on its domain, which is an open interval I.

( i ) For each point a in I, the definite integral of f from a to x considered as a function of x is an antiderivative of f. That is,…

   (ii) If F is any antiderivative of f, then for any two points (a, b) in I the definite integral of f from a to b is equal to the difference F(b) - F(a),

   由此可见，国内微积分课程的标准说法，只把上述（ii）说成是微积分基本定理的严重片面性。实际上，（i）的证明才是最困难的。

袁萌  3月13日