Example 5.40.15F5EC436555E6

Example 5.40.15.F5EC436555E6 At every point $p=(0,w_0)\in \partial B$ with $|w_0|=1$, the cluster set of $G_ m$ at $p$, that is, the set of limits of $G_ m$ along sequences in $B$ converging to $p$, is all of $[-\infty ,0]$. Indeed, set $w_ s=(1-s^2)^{1/2}w_0$ for $0<s<1$. For every fixed $0<a<1$, the points $(as,w_ s)$ belong to $B$, tend to $p$ as $s\to 0$, and satisfy $G_ m=m\log a^2$. Choosing $a=a_ s<1$ with $a_ s\to 1$ gives the limit zero, whereas the points $(s^2,w_ s)$ give $G_ m=m\log s^2\to -\infty $. These realize every value in $[-\infty ,0]$, and negativity excludes all others. Thus $G_ m$, although continuous on $B$, has no continuous extension to any point of $\partial B$ in the closure of $D$.