Can we find two functions $f_1(x)$ and $f_2(y)$ that $f_1(x)+f_2(y)=\frac{1}{x+y}$, for all independent $x$ and $y$?

Question

I wish someone can help to make the following problem clear:

Let $x$ and $y$ are independent variables.

The task is to find (or to prove it is impossible to find) two functions $f_1(x)$ and $f_2(y)$ that for all $x,y\in R$, it holds that $f_1(x)+f_2(y)=\frac{1}{x+y}$.

Intuitively, I think it is impossible, but I don't know how to formally prove it.

Thanks a lot.

Answer 1

If this is possible then we must have that:$$f_1(x)+f_2(0)=\dfrac{1}{x}$$and$$f_1(x)+f_2(1)=\dfrac{1}{x+1}$$therefore$$f_1(x)=\dfrac{1}{x}-f_2(0)=\dfrac{1}{x+1}-f_2(1)$$which implies$$\dfrac{1}{x(x+1)}=f_2(0)-f_2(1)=C$$which is a constant for every $x$ and this is a contradiction. So it is impossible.