  We study continuity of the roots of nonmonic polynomials as a function of
their coefficients using only the most elementary results from an introductory
course in real analysis and the theory of single variable polynomials. Our
approach gives both qualitative and quantitative results in the case that the
degree of the unperturbed polynomial can change under a perturbation of its
coefficients, a case that naturally occurs, for instance, in stability theory
of polynomials, singular perturbation theory, or in the perturbation theory for
generalized eigenvalue problems. An application of our results in multivariate
stability theory is provided which is important in, for example, the study of
hyperbolic polynomials or realizability and synthesis problems in passive
electrical network theory, and will be of general interest to mathematicians as
well as physicists and engineers.
