Can we define the limit using only rational nos.?

Question

Does the epsilon-delta definition require that the function $f$ is defined to map from $\mathbb R \rightarrow \mathbb R$? Would it be the same to define $f: \mathbb Q \rightarrow \mathbb Q$?

Answer 1

The concept of limit is defined for any topological space (i.e. a set of sets that are called open and satisfy the properties you'd expect from open sets. This is a very generalized representation of space). The epsilon-delta definition works for any metric space (i.e. a set of points with a function to measure distances between those points in a way that makes sense). So, yes, a limit is defined for $\mathbb{Q}$ as well for a large number of other objects.