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

I want to prove the following claim:

If $f:(0,\infty)\to\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ differentiable on $x_0=1$, then $f$ differentiable for all $x_0>0$.

Thank you.

Answer

Let $y=1+h/x$. Then

$$f'(x)=\lim\limits_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim\limits_{h\to 0}\frac{f(xy)-f(x)}{h}=\lim\limits_{h\to 0}\frac{f(y)}{h}=\frac{1}{x}\lim\limits_{h\to 0}\frac{f(1+h/x)}{h/x}=\frac{f'(1)}{x}.$$