By a theorem in Hoffmans book we know that a Blaschke product $B(z)$ is analytic in the closed unit disc everywhere except the compact set $K$ which consists of the accumulation of it's zeros. However can continuity exist at the accumulation of it's zeros? (i.e is it possible for a Blaschke product to have unrestricted limit equal to zero on the boundary?)
Thanks for any help
A Blaschke product converges on the unit circle. Now I am not sure how you want to extend your function but it will not change a lot.If you just want to extend them by continuity the maximum modulus theorem states that for an analytic function $f$ on a region $G\subseteq \mathbb{C}$ that is continous on $\overline{G}$ $$\max|f(\overline{G})| = \max|f(\partial G)|$$
Thus if $B$ extends as $0$ on the boundary of the circle it will also have to be $0$ on the circle.