Suppose we have the specrum function $\sigma :A\to \Bbb C$,$A$ is a Banach algebra,then $\sigma(x)$ is upper semi-continuous ?
We need to prove $V=\{x\in A:\sigma(x)<k\}$ is open for any $k\in \Bbb C$.For any $x\in V$,how to show that exists an open ball $U$ containing $x$ such that $U\subset V$?