设函数f（X）对任何X,Y恒有f(x+y)=f(x)+f(y),且f'(0)=2,则f'(x)=

Question

设函数f（X）对任何X,Y恒有f(x+y)=f(x)+f(y),且f'(0)=2,则f'(x)=

Answer 1

由条件知道
f(0+0)=f(0)+f(0),得f(0)=0
f'(x)=lim(h->0)[(f(x+h)-f(x))/h]
=lim(h->0)[(f(x)+f(h)-f(x))/h]
=lim(h->0)[f(h)/h]
=lim(h->0)[(f(h)-f(0))/h]
=f'(0)
=2