Function F(x,y) which is high when (both x and y are high) and (x is close to y)

Question

Given two positive real numbers $x$ and $y$, I'm looking for a function $F(x,y)$ that has the following behaviour: the higher $x$, $y$ values and the smaller $|x-y|$, the higher the number $F(x,y)$ is.

How can I define such a function ?

Answer 1

How about $\;F(x, y) = \sqrt{xy}\;$ defined on $\mathbb R_{\gt 0}\times \mathbb R_{\gt 0}$.

Indeed, on this same domain, look at $f(x, y) = xy$.

See, for example, the first octant in the graph of $F(x, y)$ below:

Answer 2

I'm maybe thinking $$F(x,y)=\frac{xy}{|x-y|}$$ might work nicely.