Basically, I have some function $f(x)$ and I would like to figure out which integer values of $x$ make it such that $f(x)$ is also an integer. I know that I could use brute force and try all integer values of $x$ in the domain, but I want to analyze functions with large (possibly infinite) domains so I would like an analytical way to determine the values of $x$.
The function itself will always be well-behaved and inversely proportional to the variable. The domain will be restricted to the positive real axis.
I thought about functions like the Dirac delta function but that only seemed to push the issue one step further back. I get the feeling that I am either going to be told that there is no way to easily determine this, or that I am misunderstanding something fundamental about functions, but I thought I'd let you all get a crack at it at least.