Consider the map $f:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$. When is it the case that there exist functions $g:\mathbb{R}^n \rightarrow \mathbb{R}^n$ and $h:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\forall x,y\in\mathbb{R}^n$, $f(x,y)=g(x)+h(y)$?

Thank you!

-
A sufficient-but-not-necessary condition is given by requiring $f$ to be an additive homomorphism, or, even more stringent, a linear map. But I suspect you're looking for something a bit broader. –  Brett Frankel Oct 29 '12 at 17:40

Iff the following two mappings are constant for all fixed $x,x_1,y,y_1$: $$t\mapsto f(x_1,t)-f(x,t)$$ $$t\mapsto f(t,y_1)-f(t,y) .$$

-