Consider a monic polynomial $$p(\lambda,a)=\lambda^n+a_1\lambda^{n-1}+\dots+a_{n-1}\lambda^{n-1}+a_n$$ with $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. Suppose that $p(\hat{a},\hat{\lambda}_i)=0$, $i=1,\dots,n$ for some $\hat{a}\in\mathbb{R}^n$ and $\hat{\lambda}_i\in \mathbb{C}$. Do there exist continuous functions $f_i$ mapping neighborhood $\mathcal{N}$ of $\hat{a}$ to $\mathbb{C}$ such that $f_i(\hat{a})=\hat{\lambda}_i$ and $p(a,f_i(a))=0$, $i=1,\dots,n$, for all $a\in \mathcal{N}$?
This question is raised from the following context: Given a real symmetric matrix $S(x)$ which is a continuous function of vector $x\in \mathbb{R}^n$. Suppose that the eigenvalues of $S(\hat{x})$ are $\hat{\lambda}_i$, $i=1,\dots,n$. I would like to know whether there are continuous functions $g_i$ relating $x$ to $\lambda_i$ with $g_i(\hat{x})=\hat{\lambda}_i$.