I have the functional equation $$ f(x+g(x)y)=f(x)+f(g(0)y)-f(0) $$ where

- $f$ is positive, monotone increasing and continuous
- $g$ is continuous and positive
- The domain of both functions is a closed interval that includes 0.

The obvious solutions are:

- $g$ constant and $f$ linear
- $f$ constant and $g$ arbitrary.

There is also another solution:

- $g(x)=x+1$, and $f(x)=ln(x+1)$.

The question is whether there are other solutions? In particular, I am interested in the question: if $g(0)\neq 1$ is $f$ necessarily linear?

Thanks.