Let $$P_{t}(z) =a_{0}(t) + a_{1}(t)z + ...+a_{n}(t)z^n$$ be a polynomial where the coefficients depend continuously on a parameter $t \in (−1, 1)$.
Assume that there exists $\text{t}_{0} \in (−1, 1)$ such that $\text{t}_{0} \neq 0$ and the roots of $P_{t}(z)$ are distinct (no multiple roots). Show that the same is true for $P_{t}$ when $t$ is sufficiently close to $\text{t}_{0}$.
I am thinking of applying Riesz representation lemma, but I am still looking for an expanded form solution.