Given a polynomial $P(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$, let the sensitivity be the largest number $\epsilon$ such that for all $-\epsilon < c < \epsilon$, we have that $P(x) + c$ has the same number of real roots as $P(x)$.
Can we give an upper bound of $\epsilon$ in terms of the coefficients?
I think the bounds of $\epsilon$ occur when $P(x) \pm \epsilon$ has a double root. So, I need a way to bound what values of $\epsilon$ create a double root.