Question

Limits and continuity of real valued functions in calculus

Answer 1

Let FF be a real valued function defined on a subset EE of RR. We say that FF is continuous at a point x∈Ex∈E iff for each ϵ>0ϵ>0, there is a δ>0δ>0, such that if x′∈Ex′∈E and |x′−x|<δ|x′−x|<δ, then |f(x′)−f(x)|<ϵ|f(x′)−f(x)|<ϵ.

This definition is taken straight out of Royden-Fitzpatrick Real Analysis.

My question is more related to the intuition behind this definition:

If I take the following function FF defined on the natural numbers(which are a subset of RR of course), for which F(x)=xF(x)=x, that is, the identity on the natural numbers but that treats them as a subset of RR.

Now this function is continous at every point of NN. If we take ϵ≤1ϵ≤1, then we can always take some δ<1δ<1. If we take an ϵ>1ϵ>1, then we can always take a respective δ<ϵδ<ϵ, but δ>δ> the largest natural number smaller than epsilon. So, indeed this function is continuous. Is this supposed to happen, and can't we somehow use this definition to show continuity of functions which are intuitively discontinuous at some points.

Is the key element here that we say that the function is continuous/discontinuous at xx as a point of a specific set EE?