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 alone, 2 alone, or both together 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 $m < n, r < s$ can be used.)