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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!

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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
up vote 3 down vote accepted

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

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Thank you! Is there perhaps some reference, to read more about this? – Kenny LJ Oct 29 '12 at 19:48
I was just guessing. Please check that it indeed works.. :) – Berci Oct 30 '12 at 11:12

I don't have access to the article, but A sufficient condition for additively separable functions, looks like it answers your question in a general form:

This paper presents a set of sufficient conditions under which a completely separable function on an open S⊂R^N is additively separable. The new condition is that different connected elements of intersections of parallel-to-the-axes hyperplanes with the domain S are intersected by common indifference surfaces.

though I'm not sure if the restriction to a "completely separable function" is important to your question without reading the article.

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