Wednesday, 9 January 2019

calculus - Functional equation f(xy)=f(x)+f(y) and differentiability




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)=limh0f(x+h)f(x)h=limh0f(xy)f(x)h=limh0f(y)h=1xlimh0f(1+h/x)h/x=f(1)x.


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