Is there a way to write $1/(x-y) = f(x) g(y)$?

Question

For $x>0$ and $y>0$, do there exist some single-variable functions $f$ and $g$ for which $$ \frac{1}{x-y} = f(x) g(y) \ ? $$

Answer 1

There isn't. Let $d=x-y$. Then $$\frac1d = f(x)g(x-d).$$

Fix $x$ and let $y$ approach $x$ so that $d$ approaches zero. The left side goes to infinity. On the right side, the factor $f(x)$ is constant. For the product to go to infinity, we must have $$\lim_{d\to 0} g(x-d) = \infty.$$

But this last equation holds for any fixed value of $x$. This is impossible.

The argument for $\frac1{x+y}$ will be essentially the same.