Added the continuous functional calculus.

src/op/banach/invertible.tex

@@ -52,6 +52,15 @@
     (3): Since the inversion map is locally a power series, it is $C^\infty$ by \autoref{theorem:termwise-differentiation}. 
 \end{proof}
 
+\begin{corollary}
+\label{corollary:invertible-boundary-explode}
+    Let $A$ be a unital Banach algebra, $x \in A \setminus G(A)$, and $r > 0$, then $\normn{y^{-1}}_A > 1/r$ for all $y \in B(x, r) \cap G(A)$. 
+\end{corollary}
+\begin{proof}
+    Let $y \in B(x, r)$, then $B(y, \normn{y^{-1}}_A^{-1}) \subset G(A)$ by \autoref{proposition:banach-algebra-inverse}. Since $x \not\in G(A)$, $r > \norm{x - y}_A \ge \normn{y^{-1}}_A^{-1}$, and $1/r < \normn{y^{-1}}_A$. 
+\end{proof}
+
+
 \begin{proposition}
 \label{proposition:swap-invertible}
     Let $A$ be a unital Banach algebra and $x, y \in A$, then $1 - xy \in G(A)$ if and only if $1 - yx \in G(A)$. 
@@ -74,5 +83,3 @@
     \end{align*}
     
 \end{proof}
-
-