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, limx→1f(x)=0, but how can I use this to prove continuity of f for every x∈R?
Any help would appreciated. Thanks
Answer
Give x0>0,
f(x)−f(x0)=f(x0⋅xx0)−f(x0)=f(xx0),
by f is continuous at x=1, when x→x0, xx0→1, then
limx→x0f(x)=f(x0).
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