Conditions for f(x,y) to coincide with some g(x) + h(y) Given a function of two variables, say f(x,y), how to know/check whether it can be simplified into some g(x) + h(y)?


Some property of some condition that condition that f(x,y) satisfies to know this.
A necessary and sufficient condition is that, for every $(x,y,z,t)$,
$$
f(x,y)+f(z,t)=f(x,t)+f(z,y).
$$
For purposes of analysis proofs, and algebraicly, I think the most useful way of formulating that property is how you already wrote it, i.e. that it can be written as the sum of two functions, one of which doesn't depend on $x$ and the other of which doesn't depend on $y$.

Assuming you're looking for a different formulation, I don't know one in general, but for differentiable $f$
$$
\frac{\partial f}{\partial x}(x,y) \text{ is independent of } y
$$
$$
\frac{\partial f}{\partial y}(x,y) \text{ is independent of } x
$$

Also, if $f$ is twice differentiable, and the second derivative is continuous, a nice formulation is the condition that
$$
\frac{\partial f}{\partial x \partial y} (x,y) = 0.
$$

(Can you prove that the above is both necessary and sufficient in the continuous, twice differentiable case?)