Continuity of the Roots of a Polynomial

Abstract

We provide a self-contained topological proof of the continuous dependence of the roots of a polynomial on its coefficients. We present it as a homeomorphism between the space of monic complex polynomials of degree n and the space of unordered n-tuples of complex numbers. Our proof uses bounds on the roots of polynomials to show that the map that associates to each monic polynomial its roots has an inverse that is a proper map.