Question

工数微积分问题 设f(0)=0 ,lim[h->0] (f(2h)-f(h))/h存在 是否是f(x)在x=0可导的充要

Answer 1

是.先证明充分性：∵ f(0)=0 ,lim[h->0] (f(2h)-f(h))/h存在,∴ lim[h->0] (f(2h)-f(0)-f(h)+f(0))/h存在存在,而且 lim[h->0] (f(2h)-f(0)-f(h)+f(0))/h= 2lim[h->0]f(2h)-f(0)/2h-lim[h->0]f(h)+f(0)/h= 2f`(0)-f`(0)=f`(0) ∴f(x)在x=0可导.

再证明必要性：∵f(x)在x=0可导,∴2f`(0)-f`(0)=f`(0)= 2lim[h->0]f(2h)-f(0)/2h-lim[h->0]f(h)+f(0)/h= lim[h->0] (f(2h)-f(0)-f(h)+f(0))/h (用h替换x).又∵f(0)=0,∴lim[h->0] (f(2h)-f(0)-f(h)+f(0))/h = lim[h->0] (f(2h)-f(h))/h.证毕