I am  looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:

1) $f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.

2) $f(m,n) \le f(r,s)$ if and only if $m n \le r  s$, with $m,n,r,s$ integers $> 0$.


I would like to know if functions with properties 1 and/or 2 exist, and if they have been studied and given a specific name (for further investigation). In particular, I would be curious to know if relatively simple closed-form functions satisfying both conditions exist, or can exist at all. (If useful to restrict the scope, the further constraints $m \le n, r \le s$ can be used.)