Let f (x) be defined for all x > 0 and be continuous. Let f (x) satisfy $\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{y}\right)$ for all x, y and f (e) = 1. Then

Taking f (x) = log x, we see that
$\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(\mathrm{y}\right)$
Clearly, f (x) is not bounded
$\text{ and }\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=-\log \mathrm{x}\to \mathrm{\infty }\text{ as }\mathrm{x}\to 0$
Also, x f (x) = x log x → 0 as x → 0
The correct option is (3) and (4)