处处连续且仅在一点可导的函数在微积分教学中的应用  

Using Function of Continuous Everywhere but Differentiable Only at One Point in Teaching Calculus

作　　者：姚卫红 YAO Weihong(School of Mathematical Sciences,Shanghai Jiao Tong University,Minhang District,Shanghai 200240,PRC)

机构地区：[1]上海交通大学数学科学学院,上海市闵行区200240

出　　处：《高等数学研究》2021年第6期4-5,3,共3页Studies in College Mathematics

摘　　要：本文利用处处连续而处处不可导的Weierstrass函数W(x)构造一个仅在x=0处n次可导其余处n-1次可导的函数.用此函数解惑在证明带皮亚诺余项的泰勒公式时,何以最后一次不能继续使用洛必达法则.In this paper,we use the Weierstrass function,a function which is continuous everywhere but nowhere differentiable,to construct a function which is but is n-th order differentiable only at x=0.With this function,we explain why L’Hopital’s rule can’t be used at the last step of the proof of Taylor’s formula with Peano remainder.

关 键 词：泰勒定理 皮亚诺余项 洛必达法则 WEIERSTRASS函数 

分 类 号：O172[理学—数学]