Question

设函数f(x)对任意实数x1,x2,横有f(x1 x2)=f(x1)f(x2),且f'(0)=1，试

设函数f(x)对任意实数x1,x2,横有f(x1 x2)=f(x1)f(x2),且f'(0)=1，试证f'（x）=f（x）。

Answer 1

f(x1+x2)=f(x1)f(x2)

令x1=x2=0
那么f(0)=f(0)*f(0)
∴f(0)=1
【若f(0)=0那么f(x1）=f(x1)*f(0)=0
则f(x)恒为0了】
∵f'(0)=1
∴f'(0)=lim(h-->0)[f(h)-f(0)]/h
=lim(h-->0)[f(h)-1]/h=1
∴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(x)[f(h)-1]/h
=f(x)*1
=f(x)