[Ericson2314]

1. You are talking about partial functions, that is a separate concern.

2. Behold https://en.wikipedia.org/wiki/Discrete_space . Topologies define continuinity, and here is a discrete topology.

3. With e.g. probability measures / expected values, which unify "discrete" and "continuous" statistics, you'll notice that there's lots of rules that are trivially obeyed in the discrete case, but take some care in the "continuous" case. For example, not ever set can have a measure in the latter but can in the former. This directly relates to discrete things being trivial to deem continuous. It's also a useful to define coarser topologies / event sigma-algebras in the finite case to better understand the issues are the unavoidable in the infinite cases. We only make the discrete discontinuous in that last "artificial" exercise.