functional equations - Solutions to $g(ab) = ag(b) + bg(a)$ - "Zero function question"

This question was recently asked and then voluntarily removed by its author. I thought it was interesting enough to get an answer. The question was as follows:

"The function $g:\mathbb R\to\mathbb R$ satisfies $g(ab)=ag(b)+bg(a)$"

For this equation, am I right in assuming that the function , $g$, is a zero function?

I'm trying to prove that any input of $g$ would equal zero, say $g(1)=0$. I can only see this working if I assume, or prove that $g$ is a zero function.

The comments pointed out that the cases $a=b=0$ and $a=b=1$ immediately give us $g(0)=g(1)=0$. However, that's not the end of the story...

Answer

Yes, there is a nonzero solution: For any constant $c$, let $g(x)=cx\log |x|$ for $x\neq 0$ and $g(0)=0$.

If $g$ is not required to be continuous, then I believe there are more solutions of the form $g(x)=x\cdot f\left(\log|x|\right)$, where $f$ is a solution to Cauchy's functional equation. I haven't worked out this part, but it's something to go on.