The aim of this note is to give a simple topological proof of the well-known result concerning continuity of roots of polynomials. We also consider a more general case with polynomials of a higher degree approaching a given polynomial. We then examine the continuous dependence of solutions of linear differential equations with constant coefficients.

We classify all polynomials $P(X)\in \mathbb{Q}[X]$ with rational coefficients having the property that the quotient $(\unicode[STIX]{x1D706}_{i}-\unicode[STIX]{x1D706}_{j})/(\unicode[STIX]{x1D706}_{k}-\unicode[STIX]{x1D706}_{\ell })$ is a rational number for all quadruples of roots $(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D706}_{j},\unicode[STIX]{x1D706}_{k},\unicode[STIX]{x1D706}_{\ell })$ with $\unicode[STIX]{x1D706}_{k}\neq \unicode[STIX]{x1D706}_{\ell }$.