So far I have shown that the equality only holds when $x=0$ or $y=0$. I also have found out that $f(nx) \leq f(x)^n$ for natural $n$ (otherwise the summation doesn't make sense).
But I do not know how to deduce the continuity. My idea is to show it with the sequence criterion for continuity, but the only thing I've shown with it is that for $n \rightarrow 0$ the function $f(nx) \rightarrow f(0)=1$, which doesn't help a lot.
I also could use that $f$ is continuous in $x=0$ to make any $f(x)$ a product of $f(1)$, but it only holds for natural numbers, thus not leading to a solution.
I would appreciate hints, not solutions. Thank you for help.