It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map T : Cn -> Cn.
Broughton, Sean A., "Continuous Dependence of Solutions of Equations on Parameters" (2014). Mathematical Sciences Technical Reports (MSTR). 107.