Thursday, 17 October 2019

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




Prove that if f:(0,)R satisfying f(xy)=f(x)+f(y), and if f is continuous at x=1, then f is continuous for x>0.





I let x=1 and I find that f(x)=f(x)+f(1) which implies that f(1)=0. So, limx1f(x)=0, but how can I use this to prove continuity of f for every xR?



Any help would appreciated. Thanks


Answer



Give x0>0,
f(x)f(x0)=f(x0xx0)f(x0)=f(xx0),


by f is continuous at x=1, when xx0, xx01, then
limxx0f(x)=f(x0).


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