Corollary 1

Let $f\in \mathscr{O}$(Sp$_A(x)$) be a germ of a holomorphic function in a neighbourhood of Sp$_A(x)$ and let $\widetilde{f}\in \mathscr{C}$(Sp$_A(x)$) be the continuous function on Sp$_A(x)$ associated with $f$. Then $\widetilde{f}(x) =f(x)$, where $f(x)$ is the element of A given by the holomorphic functional calculus.

In fact, the mapping $f\mapsto \widetilde{f}(x)$ is a continuous unital morphism from $\mathscr{O}$(Sp$_A(x)$) into A which maps the germ of the identity function in a neighbourhood of Sp$_A(x)$ to $x$. The result is then a consequence of theorem 5 of I, p. 74.