$F(x)+G(y)= e^{x+y}?$ Are there functions $F(x)$, $G(y)$, such that
 $F(x)+G(y)=e^{x+y}$ , where $x,y$ are real numbers? I have been trying all elementary functions, and have no clues on what else I could do. 

If that were the case, you would have for all $y$, $$F(1) - F(0)=F(1)+G(y) -(F(0)+G(y)) = e^{1+y}-e^y=e^y(e-1),$$ so $e^{y}(e-1)$ would be constant.

Motivation: Initially I thought of taking a partial derivative, but with more generality, it suffices to take a difference $f(0+1,y) -f(0,y)$.  
