I want to prove the following claim:
If f:(0,∞)→R satisfying f(xy)=f(x)+f(y), and if f differentiable on x0=1, then f differentiable for all x0>0.
Thank you.
Answer
Let y=1+h/x. Then
f′(x)=limh→0f(x+h)−f(x)h=limh→0f(xy)−f(x)h=limh→0f(y)h=1xlimh→0f(1+h/x)h/x=f′(1)x.
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