So I have a function $$f: \mathbb{Q} \rightarrow \mathbb{R}, f(x)=x$$ and need to state with justification whether or not it is continuous. I seem to be having trouble actually interpreting the domain of the function.
If I was to draw the graph of $f(x)$ there would be a 'hole' wherever x is irrational, which would suggest discontinuity to me. However there are no irrationals in the domain of $f$ and so it doesn't actually have any points of discontinuity, and I want to say $f$ is continuous on $\mathbb{Q}$. How should I be interpreting this? And should I use an epsilon-delta argument or a limit argument?
Use the definition of continuity that for every sequence $ \{ x_n \} _ { n \ge 1}$ converging to a given $x \in \mathbb{R}$, the image sequence $f(x_n) \to f(x)$
Note: You can use the $\epsilon - \delta$ definition only when there exists some $r > 0$ such that $ ( x - r, x+ r ) \subset \mathbb{domain}(f)$