In complex analysis we have
Mergelyan's theorem :
Let $K$ be a compact subset of the complex plane $C$ such that $C∖K$ is connected. Then, every continuous function $f : K \to C$, such that the restriction $f \to int(K)$ is holomorphic, can be approximated uniformly on $K$ with polynomials. Here, $int(K)$ denotes the interior of $K$. Mergelyan's theorem also holds for open Riemann surfaces.
But I wonder, why does this fail for several complex variables? How to prove that it can fail in higher dimensions (spaces of several complex variables) and what are some examples of such failures ?