  Sufficient and necessary conditions on the spectral measure of a self-adjoint
operator $A$, acting in a Hilbert space, are obtained, under which for any
continuous scalar function the operator function $\phi(A+\gamma B)$ is
holomorphic with rspect to the coupling constant $\gamma$ in a neighborhood of
$\gamma=0$, where $B$ is a self-adjoint operator. The sharpest results are
obtained in the case where $B$ is a rank-one operator.
