Consider the functional equation $f(y,z)g(y+z)=h(y)$, where the functions $f,g,h$ are all continuously differentiable and where $h$ is not constant.
I have just about convinced myself that any solution to this must take the form $g(x)=e^{cx}$ for some constant $c$ and $f(y,z)=\tilde f(y)e^{-cz}$, for some function $\tilde f$. But I don't have a proper proof.
I suspect the proof is so obvious (except to me) that it's unworthy of being documented in, e.g., Aczel-Dhombres, or else it's a well-known result (again, except to me). Or maybe there's another solution I'm missing?
A pointer to an existing result or proof or counterexample would all be welcome!