函数在一点可导的等价形式

An Equivalent Definition for Derivative

通过对一道考研试题的推广,得到函数在某点的可导的一个等价形式,即若函数f(x)在x=0处连续,且f(0)=0,limx→0[f(x)-af(bx)/x]=K,其中0<|ab|≠1,0<|b|≤1,且f(x)在x=0处满足Lipschitz条件,则有f′(0)=K/(1+ab).

In this paper, the following equivalent definition for derivative is obtained. If a function f（x） is continuous at x = 0, satisfies Lipschitz condition at x = 0, and f（0） =0 , lim x→0 f（x）-af（bx）/x =K.where 0〈｜ab｜≠1,0〈｜b｜≤1,then f′（0）= k/1＋ab.