So I want to prove that if we have a polynomial in the form of
$f(t,z)=\sum_{i=0}^n a_i(t) z^i$
where the coefficients $a_i(t)$ are continuous w.r.t. $t$, there then exists a continuous function $r_k(t)$ such that
$f(t,r_k(t))=0$,
or in other words such that at least one of the roots of $f(t,z)$ is continuous w.r.t. $t$.
I think I understand proofs of continuity like in:
which show through Rouche that for a given polynomial which has a minimum magnitude of the difference between its roots as $\tilde{r}$ and for any $0<\epsilon<\frac{\tilde{r}}{2}$, we will have that there exists a $\delta>0$ s.t. if we perturb the coefficients by any values less than or equal to $\delta$ we will have that the roots of the resulting perturbed polynomial will be within $\epsilon$ of the original roots (magnitude wise).
I can see how this is pretty much the definition of continuity, but the upper bound on $\epsilon$ is confusing to me. Isn't it possible that the upper bound on this $\epsilon$ gets arbitrarily small as we perturb the coefficients and roots begin to approach each other? Also, how could we prove an intermediate value theorem for the roots with this upper limit on $\epsilon$? Sorry if these are foolish questions.
$\textbf{EDIT}$: oh I just realized that if the above is true for some $\epsilon$, then it is obviously true for any larger value of $\epsilon$. Also, looking at the Rouche proof, I think that $\epsilon$ just has to be such that the balls of radius $\epsilon$ about the roots of the original polynomial do not contain other roots on their boundary, so I think I can just consider a larger $\epsilon$ that bounds roots that are becoming arbitrarily close together in order for the resulting $\delta$ to be well defined even as roots potentially become identical as we perturb the coefficients, and I think that pretty much gives me a straightforward $\epsilon-\delta$ continuous relation between the coefficients and resulting roots. I am still not sure though, but I think the above can give me the result for any $\epsilon>0$ where I can consider $0<\epsilon_1<\frac{\tilde{r}_1}{2}$ as in the original proof to get a corresponding $\delta(\epsilon_1)$, and this $\delta(\epsilon_1)$ also covers the case for $\hat{\epsilon}_1=\frac{\tilde{r}_1}{2} $. Then I can consider $\frac{\tilde{r}_1}{2}<\epsilon_2<\frac{\tilde{r}_2}{2}$ where $\tilde{r}_2$ is the next minimal difference in the polynomial roots. Since this $\epsilon_2$ is such that all of the balls centered on the roots still do not have any roots on their boundaries, I am pretty sure I can still use the linked Rouche proof to obtain a suitable $\delta(\epsilon_2)$. Continuing in this manner for all the possible root differences and beyond, I think I can arrive at the familiar "for all $\epsilon>0$" definition of continuity even if multiple roots begin to converge towards being identical.
