Question

设f(x)，对任意实数x1,x2，有f(x1+x2)=f(x1)+f(x2)且f'(0)=1,求证 f'(x)=f(x)

Answer 1

f(x)对任意实数x1,x2,有f(x1+x2)=f(x1)+f(x2),
取x2=0,f(x1)=f(x1)+f(0)
所以f(0)=1
所以利用导数的定义公式
f'(x)=lim(h->0)[f(x+h)-f(x)]/h
=lim(h->0)[f(x))f(h)-f(x)]/h
=f(x)lim(h->0)[f(h)-1]/h
=f(x)lim(h->0)[f(h)-f(0)]/h
=f(x)*f'(0)
=f(x)
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