Pointwise polynomial approximation of a half-plane sign

There is a sequence of complex polynomials converging pointwise on the whole plane to the sign of the imaginary part, with value zero on the real axis. For each $n$, take the compact union of the upper rectangle $[-n,n]+i[1/n,n]$, its lower reflection and the real segment $[-n,n]$. Its complement is connected. The function equal to one, minus one and zero on disjoint neighborhoods of the three components is holomorphic. The polynomial Runge theorem provides a polynomial within $1/n$ of it on the union. Every fixed point lies in its corresponding component for all large $n$, proving pointwise convergence. The limit is discontinuous, so the convergence cannot be locally uniform near the real axis. This illustrates how pointwise approximation by holomorphic functions differs from locally uniform approximation.