Suppose that $E \subset \mathbb{R}$ is compact and $m(E)>0$. Let $\Omega=\mathbb{C} \setminus E$ and \begin{equation} f(z)=\int_E \frac{dt}{t-z} \end{equation} for all $z \in \Omega$. I think that $f$ cannot be extended to an entire function because as $z \to \max E$ and $z>\max E$, we have $\frac{1}{t-z} \to -\infty$. Is my observation correct? Anyone can provide the details?
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Hint: $f(z)\to0$ as $|z|\to\infty$. Now what kind of contradiction do you get if $f$ is assumed to be entire, in view of the Liouville's Theorem? – Sangchul Lee Jun 02 '20 at 11:42
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Maximum and minimum may be isolated points of $E$. So your idea doesn't work. – Kavi Rama Murthy Jun 02 '20 at 11:55
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I see the point now. I really miss the use of Liouville's Theorem. – KK Kwok Jun 02 '20 at 12:03