Can $f(z)=\int_E \frac{dt}{t-z}$ be extended to an entire function?

Question

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?