Holomorphic spectral mapping theorem ($\sigma (f(x))=f(\sigma (x))$)

For a holomorphic function near the spectrum, the spectrum of its functional-calculus value is exactly the image of the original spectrum. If a scalar is absent from that image, the reciprocal of the shifted function is holomorphic on a smaller spectral neighborhood and supplies an inverse. If it is attained at $\lambda$, a divided difference factors $f(x)-f(\lambda )$ through $x-\lambda$. These factors commute, so invertibility of their product would contradict $\lambda \in \sigma (x)$.