Are the real root of a polynomial continuous to its real coefficients?

Question

I have the following question.

Given a polynomial with real coefficients, are the real root of the polynomial continuous to the real coefficients of the polynomial when the number of real roots does not change?

In other words,

For $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0, a_i\in R$, let's denote the number of real roots of the polynomial as $N(a_n, a_{n-1}, ..., a_1, a_0)$.

Then, are the real roots of $a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ continuous to $a_i$ for all $[a_n, a_{n-1}, ..., a_1, a_0] \in R^{n+1}$ such that $N(a_n, a_{n-1}, ..., a_1, a_0)$ is constant?

I know that the roots of a polynomial are continuous to its coeffcients, when the roots and the coefficients are complex numbers. (Continuity of the roots of a polynomial in terms of its coefficients)

I also know that the real roots of a polynomial may not be continuous to its coefficients. (A counterexample would be $x^2+a$ at $a=0$, as the real root disappears when $a$ goes below $0$)

I wonder if such "disappearance" of roots is prevented, the real roots of the polynomial can be continuous in terms of the real coefficients of the polynomial?

Thank you in advance.

Answer 1

No--or at least not unless you count real roots with multiplicity. It is possible for real roots to disappear in one place and simultaneously appear elsewhere as the coefficients vary. For example, let $f(x) = x(x-1)^2(x+1)^2$, and consider the equation $f(x) = c$ as $c$ ranges over a small interval $(-\varepsilon, \varepsilon)$ around $0$. At $c = 0$, $f(x) = c$ has three real roots, $0$ and $\pm 1$, with the latter two having multiplicity $2$. When $c$ is small and positive, the double root at $1$ splits into two distinct roots, while the double root at $-1$ disappears (i.e. splits into two imaginary roots). For $c$ small and negative, the opposite happens. So if we disregard multiplicity, there are exactly three real roots for all $c$ in $(-\varepsilon, \varepsilon)$, but their locations jump at $c=0$.

I would imagine that the statement becomes true if you count roots with multiplicity, or alternatively just exclude the locus of polynomials that have repeated roots.