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$f: \mathbb{C} \to \mathbb{C}$ is continuous (here $\mathbb{C}$ is the complex plane) and $f$ is analytic everywhere on $\mathbb{C}$ except possibly on the interval $[-1,1]$. Is $f$ an entire function?

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You know Morera's theorem? – Jonas Meyer Nov 23 '12 at 8:03
This answers your question in more generality. – EuYu Nov 23 '12 at 9:00

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