On continuity of roots of a polynomial depending on a real parameter

Let $Pol_n$ denote the space of degree $n$ monic polynomials. You have the natural continuous map $R: Pol_n \to Q= {\mathbb C}^n/S_n$ sending each polynomial to its set of (unordered) roots; here $S_n$ is the permutation group on $n$ letters. Let $q: {\mathbb C}^n\to Q$ denote the quotient map. The key observation is that the map $q$ has the path-lifting property (no uniqueness of the lift is assumed, only existence). This is a special case of the path-lifting property for orbifold-coverings, Lemma 4.1.3 here, or Lemma 2 in

M. Armstrong, The fundamental group of the orbit space of a discontinuous group. Proc. Cambridge Philos. Soc. 64 (1968) 299–301.

Note that Armstrong works even in greater degree of generality, namely with proper (but not, in general, free) discrete group actions on locally compact metrizable topological spaces.

Now, we can prove the path-lifting property you are asking for. Take a map $f: {\mathbb R}\to Pol_n$, compose it with $R$. The result is a map $$ g: {\mathbb R}\to Q. $$ Applying the above path-lifting property to $g$ we obtain a lift $$ \tilde g: {\mathbb R}\to {\mathbb C}^n. $$ Let $\tilde g_1$ denote the first component of this lift (the "1st root" of the polynomial $f(t)$). Then the map $$ \tilde f: t\mapsto (f(t), \tilde g_1(t))\in Pol_n \times {\mathbb C} $$ is the lift you want.

Edit: Armstrong's proof depends on Theorem 2 (path lifting property for "open light maps") in

E.E. Floyd, "Some characterizations of interior maps". Ann. of Math. 51 (1950), 571-575.

Armstrong simply observes that quotient maps like $q$ in your case, are "light and open": Open is clear (since it is a quotient map), "light" follows from the fact that point preimages are discrete (finite in your case).

Hence, Floyd's theorem applies. (One needs a tiny compactness argument since Floyd assumes that the domain and the range are compact, but the path-lifting property is a purely local issue, so it works for locally compact spaces.)

Floyd's paper is available (for free) via Jstor, once you open a free account with them.