Lipschitz functions of self-adjoint operators in perturbation theory

Let $A$ be a self-adjoint operator in a Hilbert space. In order that for each differentiable function $f$ and for each self-adjoint operator $B$ one should have the estimate $\|f(B)-f(A)\|\le c_f\|B-A\|$ it is necessary and sufficient that the spectrum of the operator $A$ be a finite set. If $m$ is the number of points of the spectrum of the operator $A$, then for the constant $c_f$ one can take $8(\log_2m+2)^2[f]$, where $[f]$ is the Lipschitz constant of the function $f$.