A question on subordinate matrix norm Suppose: 


*

*$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$

*${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x  + A_0$ is a matrix polynomial,  and $x $ is a complex variable.

*${\rm q}(x) =\rm{w}_m x^m + \cdots +w_1 x + w_0$

*$U= \left\{ {x  \in \mathbb{C}:\left\| P(x)^{-1} \right\| \ge (\varepsilon q(|x|))}^{-1} \right\}$, where $\left\| \cdot \right\|$ is any subordinate matrix norm.


Why can the boundary of $U$ be written in the form $S = \left\{ x  \in \mathbb{C}:\left\| P(x )^{-1} \right\| = (\varepsilon q(|x|))^{-1} \right\}$?

A partial answer...
First a remark: in order for $U$ to be well defined, we need to suppose that $P(x)$ is invertible for all $x \in \mathbb C$. 
Now the map $$h : x \longmapsto \Vert P(x)^{-1} \Vert - (\varepsilon q(|x|))^{-1}$$ is continuous. Hence $h^{-1}((0, +\infty))$ is open and a point $x \in \mathbb C$ with $h(x) > 0$ cannot belong to the boundary of $U$ named $\partial U$. Consequently $\partial U \subset S$.
Remains to prove that $S \subset \partial U$...
