Let $\phi(x) = |1 - 1/x|$ for all $x > 0$. Then $\phi(x)$ is an unbounded
continuous map from $(0, \infty)$ onto $[0, \infty)$ which maps the set
$\mathbb {R_+} \setminus \mathbb {Q_+}$ of irrational points in $(0, \infty)$
onto itself.