Find all functions $f : \mathbb{(0,\infty)}\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$ if $f$ is continuous at $x=1$
This problem was looking quite easy at first but the domain of positive reals is posing me a problem. I couldn't plug in zero's for $x$ and $y$. I tried putting $x=y$ but the result $f(2x) = 2xf(x)$ couldn't be used as a recurrence relation $\infty$ times as that would yields $f(0)$ again. I've run out of ideas. Please help.