How do you define the continuity of real functions over Natural numbers? Consider a well known function $ f(x) = x$, we all know that the function is continue over $ \Bbb R $ but how about if define it over $\Bbb Z$ or $\Bbb N$ ? Clearly we can't use the same approach to calculate the limit because values are not included in $\Bbb N$. So is it now a discontinue function?

The usual topology of $\Bbb Z$ or $\Bbb N$ is discrete, so continuity becomes trivial.

A function from $S$ to $T$ is continuous if the preimage of an open set in $T$ is open in $S$. When $S=\Bbb N$, every set is open in $S$ because the topology is discrete, so every function is continuous.