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. In this note, we show that (1) $\phi(x)$ has {\it transitive}
points (i.e., points with dense orbits) in $(0, \infty)$ and has dense
(irrational) periodic points in $(0, \infty)$; (2) $\phi(x)$ has bounded
uncountable invariant 2-scrambled sets of irrational points in $(0, 3)$; (2)
for any countably infinite subset $X$ of points (rational or irrational) in
$(0, \infty)$, there exists a dense unbounded uncountable invariant
$\infty$-scrambled set $Y$ of irrational transitive points in $(0, \infty)$
such that, for any $x \in X$ and any $y \in Y$, we have, under the convention
that $| 1 - {\frac 1 0}| = \infty$ and ${\frac 1 \infty} = 0$, $$\limsup_{n \to
\infty} |\phi^n(x) - \phi^n(y)| = \infty \quad \text{and} \quad \liminf_{n \to
\infty} |\phi^n(x) - \phi^n(y)| = 0.$$