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, lim, but how can I use this to prove continuity of f for every x \in \mathbb R?



Any help would appreciated. Thanks


Answer



Give x_0>0,
f(x)-f(x_0)=f\left(x_0\cdot\frac{x}{x_0}\right)-f(x_0)=f\left(\frac{x}{x_0}\right),
by f is continuous at x=1, when x\to x_0, \frac{x}{x_0}\to1, then
\lim\limits_{x\to x_0}f(x)=f(x_0).


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