设f(x)是定义在(负无穷,正无穷)上的函数,f(x)≠0,f'(0)=1,且对任意x,y有f(x+y)=f(x)f(y),证明f（x）在（正无穷,负无穷）内处处可导,且f'(x)=f(x)

Question

设f(x)是定义在(负无穷,正无穷)上的函数,f(x)≠0,f'(0)=1,且对任意x,y有 f(x+y)=f(x)f(y),证明
f（x）在（正无穷,负无穷）内处处可导,且f'(x)=f(x)

Answer 1

因为f(x)≠0,所以f(0)≠0,
f(x+y)=f(x)f(y)中取x=y=0,得
f(0+0)=f(0)f(0),f(0)=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)-f(0)]/h=f(x)f′(0)=f(x)