Rank of a spectral function of a Hermitian matrix: $\operatorname{rank} f(A) = \#\{i : f(\lambda_i) \ne 0\}$

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Rank of a spectral function of a Hermitian matrix: $\operatorname{rank} f(A) = \#\{i : f(\lambda_i) \ne 0\}$ · Prove2Me