= 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.
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