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I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$.

Is this problem related to the removable singularity? and are there some other sets that could draw same conclusion?

Xaviere
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  • This is related to something called (continuous) analytic capacity. If there is no answer, I'll post something when I have more time. – mrf Dec 18 '14 at 15:49
  • One can use continuity to show that every integral of $f$ along a closed trigonal contour vanishes (wherever one crosses $[a,b]$--which happens at most twice--instead go around it, with opposite directions very nearly canceling each other out). – Jonathan Y. Dec 18 '14 at 16:03
  • See also this question for something very similar. – mrf Dec 19 '14 at 10:22

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