TITLE: Uniqueness of a holomorphic function with certain boundary values on an arc
QUESTION [4 upvotes]: Is it true that if a holomorphic function in the unit disk converges uniformly to the $0$ function some connected arc of the unit circle, this function is globally null?
If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)

REPLY [0 votes]: Suppose $f$ is analytic on the disc, whose limits at the boundary yields a set of zeros with non-empty interior, and $f$ is continuous on some neighborhood (in the closed disc) of one such interior accumulation point. By reflecting through the circular boundary we can extend $f$ to be analytic on a neighborhood (in the plane) of the accumulation point. As was indicated by a comment, this is essentially the Schwarz reflection principle, though in a slightly different form than usually stated. Thus the extension has an accumulation of zeros in its domain, and so is identically zero.
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