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on the boundary of analytic functions

I was thinking of a problem I will feel happy if you give hint to solve this. Suppose I have a function $f$ that is analytic on the unit disk $D=\{z \in \mathbb{C}:|z|<1\}$ that extends continuously to $\overline{D}$. If $f$ is identically zero on some segment of of the boundary, is it then true that $f$ is identically zero on the entire boundary?

I know that analytic functions that are zero at an accumulation point inside the domain of analyticity are identically zero throughout the entire domain, but I don't know what (if anything) can be said if something similar occurs on the boundary of the domain.

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  • Dear rita, in fact, such a function is identically zero. Since this question has already been asked (and answered, quite nicely!) on this website, I've closed it. Please see the link above. – Akhil Mathew Mar 17 '11 at 13:32
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    Hum, isn't this almost a word-for-word copy of the duplicate question listed above? – Willie Wong Mar 17 '11 at 13:39

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