Consider a function
$$ f: \Bbb{Z} \rightarrow \Bbb{Z} $$
Over the integers. Furthermore consider a number E such that there doesn't exist an integer R such that $f(R) = E$ or formally stated
$$ E | \lnot \left( \exists R| f(R) = E \right)$$
Is it ever possible that for each natural number $i$ there exists $w_i$ such that
$$ f(w_i) \equiv E \mod i $$
In other words,
$$ f^{-1}(E) \not \in \Bbb{Z}$$
Yet
$$ f^{-1}(E) \mod i \in \{{x \mod i}\} \forall i $$
This is a generalization of the question:
Do there exist Artificial Squares?
To now arbitrary functions.
Clearly for squares this is not the case, and the answer is fairly easy to generalize for any function of the form $f(x) = x^{a}$ but making a statement such as this over all possible functions seems bold.
$f(x)=1+|x|$ is never zero, but $f(|i|-1)\equiv 0\pmod i$ for all $i\neq 0$.