Let
$l(\xi)=\xi_n^k + $ lower order terms in $\xi_n$
be a polynomial in $\mathbb{C}^n$, $\xi'=(\xi_1,\ldots,\xi_{n-1})$, and $l_{\xi'}(z)=l(i\xi',iz)$. Further, let $\lambda_1(\xi'),\ldots,\lambda_k(\xi')$ denote to the zeros of $l$.
Now, why is the mapping $\xi'\mapsto \lambda_i(\xi')$ continuous for all $i=1,\ldots,k$?
edit: This is part of a proof of the Malgrange-Ehrenpreis theorem. There it vaguely refers to Rouché's theorem. If it applies to the above-mentioned problem I still don't see how.