I am studying Factorisation of $H^p$ functions from Hoffman's Banach Spaces of Analytic Functions. The author is talking about extending the Blaschke product continuously to the accumulation points of the zeroes of the Blaschke product. But the author does not exactly define what it means to extend the function continuously.
The Blaschke product is in fact analytic in $\mathbb C \setminus K$ where $K$ is as mentioned in the theorem in the attached photo. By "extending to the an accumulation point" does he mean extending from $\mathbb C \setminus K$ to the set containing $\mathbb C \setminus K$ and the accumulation point?
If that's the case, if $\alpha$ is an accumulation point of $(\alpha _n )$ then we can find a subsequence $(\alpha _ {n_k})$ which converges to $\alpha$. Thus, by continuity of $B$ on the interior, we have that $B(\alpha ) = \lim _{k} B(\alpha _{n_k}) = 0$.
On the other hand, the $\lim _{r \to 1} |B(re^{i\theta})| = 1$ for almost all $\theta$. But how does that contradict anything?
