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 allow existence, the further constraints $m < n, r < s$ can be used.



Two-variable continuous function which results in an integer if and only if arguments are integer
Such functions as you require do not exist. Your requirements impose that $f(1,1) < f(1,2) < f(2,2)$ (for example, $f(1,1) \leq f(1,2)$ by (2), but $\neg (f(1,2) \leq f(1,1))$ also by (2), so we have $f(1,1) < f(1,2)$). Now consider the continuous function $x \mapsto f(x,x)$ on $[1,2]$: by the intermediate value theorem, there exists $x$ with $1<x<2$ such that $f(x,x) = f(1,2)$. Since $f(1,2)$ is an integer but $f(x,x)$ is not by your requirement (1), this is a contradiction.


