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Suppose that $f:\mathbb{C}\to\mathbb{C}$ is continuous, and analytic everywhere except possibly the unit circle. I want to show that $f$ is in fact entire. I guess the idea is to consider the integral along the disc and argue that it must be zero by Cauchy's theorem, and ...

What is the final step here?

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Morera's theorem, and argue by continuity in the integral. –  Kris Aug 29 '12 at 0:41
    
@Kris, you should write your comment as an answer. –  DonAntonio Aug 29 '12 at 2:10

1 Answer 1

up vote 1 down vote accepted

Morera's statement says that if $f: D \to \mathbb{C}$ is continuous on some connected open set $D \subset \mathbb{C}$, and $\int_{\gamma} \ f = 0$ for any closed piecewise $C^1$ curve $\gamma \subset D$, then $f$ is holomorphic on $D$.

In this case, $D = \mathbb{C}$, and it's clear that $\int_{\gamma} \ f = 0$ for any $\gamma$ which does not intersect the unit circle. If $\gamma$ does intersect the unit circle, then we can argue by continuity of the integral that $\int_{\gamma} \ f = 0$ still (split up $\gamma$ into finitely many parts, each of which intersects the circle, and deform each 'infinitesimally' so that each part no longer intersects the unit circle).

Now we are done.

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