Is Every Holomorphic Near an Entire? Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded. Let $U$ be the interior of $K$.
Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction to $U$ is holomorphic.
Assume furthermore that for every closed curve $\gamma\subset K$, the integral $\int_\gamma f(z)dz$ vanishes. (this is only relevant if $K$ is not simply connected). If you prefer, you can assume that $K$ is simply connected.

Can $f$ be uniformly approximated by entire functions?

 A: This theorem does not follow in a straightforward way from Mergelyan's theorem, which on its face applies only to bounded domains.
There is, however, a related theorem called Arakelian's theorem which settles the matter.  For completeness:
Definition: A "hole" of a closed subset $E$ of $\mathbb{C}$ is any bounded component of the complement of $E$.
Definition A set $E$ is Arakelian if $E$ has no holes, and if for every closed disk $D$, the union of all holes of $E \cup D$ is bounded.
Theorem Let $E$ be Arakelian. If $f$ is continuous on $E$ and holomorphic on the interior of $E$, then $f$ can be uniformly approximated by entire functions.
A proof deriving this result from Mergelyan's theorem can be found here.

I would also like to remark that Mergelyan's theorem (while usually stated for polynomial approximation) actually applies to domains with holes as well, if you just allow rational approximation with "poles in the holes". 
I would also like to draw attention to how crazy Mergelyan's theorem is.  One consequence is that one can approximate an antiholomorphic function by entire functions on any set without interior.  For instance, one can approximate $\overline{z}$ on the "plus sign" consisting on the interval $[-1,1]$ on the real axis and "$[-i,i]$" on the imaginary axis.  If anyone can explicitly construct such approximating functions, I would be very happy to see them! 
A: As mentioned in the comments, this is true if $K$ is compact and the complement of $K$ in the Riemann sphere is connected : it is the content of Mergelyan's Theorem on uniform polynomial approximation of holomorphic functions.
EDIT If $K$ is only assumed to be closed, this is also true with the additional assumption that $K$ is locally connected at infinity : it is the content of Arakelian's theorem on the uniform approximation of holomorphic functions by entire functions.
It is false in general if the complement of $K$ is not connected : consider $K$ the annulus $\{1/2 \leq |z| \leq 2\}$ and $f(z):=1/z^2$. Then $f$ is continuous on $K$, holomorphic in the interior of $K$, and $\int_\gamma f(z)\, dz=0$ for all closed curve $\gamma$ contained in $K$. However, $f$ cannot be uniformly approximated on $K$ by entire functions $(f_n)$. Indeed, if it were the case, then the functions $g_n(z):=zf_n(z)$ would uniformly approximate $g(z):=zf(z)=1/z$ on $K$. This contradicts the fact that $\int_\mathbb{T} g_n(z)\, dz=0$ for all $n$, whereas $\int_\mathbb{T} g(z) \, dz = 2\pi i \neq 0$. Here $\mathbb{T}$ is the unit circle. 
