this is more of a broader question.
Say I have an analytic complex function $f$ that is defined on the open unit circle, but I know that the limit of $f$ when $|z|$ approaches 1 is 0. Can you define a function $g$ that satisfies:
for $|z| \lt 1$, $g(z) = f(z)$
for $|z| = 1$ $g(z) = 0$
And would that function remain analytic?