  Let $\phi(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $\phi(x)$ in
the usual way to become a continuous map from the compact topological (but not
metric) space $[0, \infty]$ onto itself which also maps the set of irrational
points in $(0, \infty)$ onto itself. In this note, we show that (1) on $[0,
\infty]$, $\phi(x)$ is topologically mixing, has dense irrational periodic
points, and has topological entropy $\log \lambda$, where $\lambda$ is the
unique positive zero of the polynomial $x^3 - 2x -1$; (2) $\phi(x)$ has bounded
uncountable {\it invariant} 2-scrambled sets of irrational points in $(0, 3)$;
(3) for any countably infinite set $X$ of points (rational or irrational) in
$(0, \infty)$, there exists a dense unbounded uncountable {\it 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 $\limsup_{n \to \infty}
|\phi^n(x) - \phi^n(y)| = \infty$ and $\liminf_{n \to \infty} |\phi^n(x) -
\phi^n(y)| = 0$. This demonstrates the true nature of chaos for $\phi(x)$.
