The dependence of (real) roots on the coefficients of a polynomial isn’t even continuous; an arbitrarily small change to the coefficients of a polynomial with a

The dependence of (real) roots on the coefficients of a polynomial isn’t even continuous; an arbitrarily small change to the coefficients of a polynomial with a double root will result in either the appearance of an extra root or in the (real) root disappearing entirely. In the complex plane the situation is much happier, but root-finding remains extremely poorly conditioned for examples like Wilkonson’s.