If a continuous function $g:\mathbb{R}^{+}\to\mathbb{R}$ satisfies functional equation $g(xy) = g(x) + g(y)$ for all $x, y \in \mathbb{R}^{+}$ and there is a number $z \neq 1$ such that $g(z) = 0$ then $g(x) = 0$ for all $x\in\mathbb{R}^{+}$..
- I have shown that if g(z) = 0 then as $g(z^{any rational power})$ = $zg(z)=0$ , but how can we show that from this that g(x) = 0 for all x belonging to R ?
2. And is it possible that there exists function satisfying the above functional equation but its being not continuous at some points ?