Let $f$ be a positive function finite everywhere defined in $(0, \infty)$ and $$M_f(s) = \sup_{x\in (0, \infty)} \frac{f(sx)}{f(x)} $$
Is easy to check that $M_f(x_1x_2)\leq M_f(x_1)M_f(x_2)$ and $\frac{1}{M_f(x^{-1})}\leq M_f(x)$
Then says that follows from this that if M_f is bounded in the neighborhood of 1 then it is finite everywhere.
I tried a few things but i cant get a proof of that, any help would be appreciated, thanks.