Let $f: \mathbb{Q}_0 \to \mathbb{R}: \frac {p}{q} \mapsto \frac{1}{q}, (p,q) = 1, q > 0$
Does $f$ have a continuous extension to $\mathbb{R}$?
My attempt:
I showed that $\lim_{x \to a, x \in \mathbb{Q_0}} f(x) =0$ for any $a \in \mathbb{Q}$.
Hence, $f$ is continuous nowhere.
It follows that $f$ cannot have a continuous extension, since the restriction of such an extension to $\mathbb{Q}_0$ would be continuous.
Is this correct?