Yet another proof that the roots of a polynomial depend continuously on the coefficients

TL;DR

This paper provides a concise proof demonstrating that the roots of a complex polynomial vary continuously with its coefficients, emphasizing the stability of roots under small coefficient changes.

Contribution

It offers a simple, direct proof using infinitesimals to establish the continuous dependence of polynomial roots on coefficients.

Findings

Roots depend continuously on coefficients

Infinitesimal perturbations lead to infinitesimal root changes

Proof simplifies understanding of root stability

Abstract

The roots of a complex polynomial depend continuously on the coefficients; that is, an infinitesimal perturbation of the coefficients results in an infinitesimal perturbation of the roots. A short, straightforward proof of this is possible using infinitesimals.