Ničle polinomov so zvezno odvisne od koeficientov polinomov

Vsak polinom stopnje n s kompleksnimi koeficienti ima (ob upoštevanju večkratnosti) natanko n ničel. Intuitivno se zdi, da se ob majhni spremembi koeficientov polinoma tudi njegove ničle le malo spremenijo. Ni pa povsem očitno, kako bi to dokazali, saj za polinome stopnje večje od štiri nimamo eksplicitne formule za ničle. Prav tako ni očitno, kako definirati pojem bližine na množici ničel, ki bo upošteval, da se ob spreminjanju koeficientov lahko spremeni tudi večkratnost ničel.

Roots of polynomials depend continuously on the coefficients of polynomials

Every polynomial of degree n with complex coefficients has exactly n roots (counting their multiplicity). Intuitively, it seems to be the case that if the coefficients of a polynomial are slightly altered, their roots will also differ by just a little. It is not entirely obvious, however, how one would prove that, since there is no explicit formula for roots of polynomials of degree four or more. It is also Not obvious, how one would define ``nearness’’ on a set of roots, which would consider that by changing the coefficients, their multiplicity may also change.