Let $f$ be an analytic function for a domain $D$ of $\mathbb{C}$ into a Banach algebra $A$. Suppose that, for all $\lambda \in D$, $\text{Sp}f(\lambda)$ is finite or a sequence converging to $0$. Suppose that $\mu \neq 0$ and $\mu \in \text{Sp}f(\lambda_0)$ for some $\lambda_0 \in D$.

Consider the set $E = \{ \lambda \in D: \mu \in \text{Sp}f(\lambda) \}$.

B. Aupetit mentions in a proof he writes for Theorem 3.4.26 in his book *A Primer on Spectral Theory*, that this set $E$ is closed by the upper semicontinuity of the spectrum. I am struggling to see why this is true.

Can anyone please point me in the right direction as to how I can show that $E$ is closed by the upper semicontinuity of the spectrum?