Consider a compact set $K\subset \mathbb{R}$ with positive measure (i.e. $\mu(K)>0$), and for $z\in\mathbb{C}\backslash K$, define the holomorphic function $f$ on $\mathbb{C}\backslash K$ by \begin{equation} f(z):=\int_K \frac{dt}{t-z}. \end{equation}
Now the question is can $f$ be extended to an entire function? Clearly $f(z)\to 0$ as $z\to \infty$ in every direction, and I'm not familiar with entire functions of this kind, this seems to me a realy pointless question :(
Choose $R > 0$ such that $K$ is contained in the disk with center at the origin and radius $R$. For $|z| > 2R$ and $t \in K$ is $$ |t-z| \ge |z| - \frac 12|z| =\frac 12 |z| $$ and therefore $|f(z) | \le \frac {2}{|z|} \mu(K)$ for $|z| > 2R$.
This shows that $$ \lim_{z \to \infty} f(z) = 0 \, , $$ which is a stronger statement than being “bounded in every direction $z \to \infty$.”
If $f$ could be extended to an entire function then that function would be bounded on $\Bbb C$, and therefore constant (Liouville's theorem), and that constant is necessarily zero.