  In this article we prove that for $p>2$, there exist pairs of self-adjoint
operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in
the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the
differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class
$\boldsymbol{S}_p$ but $f(A_2,B_2)-f(A_1,B_1)\not\in\boldsymbol{S}_p$. A
similar result holds for functions of contractions. We also obtain an analog of
this result in the case of triples of self-adjoint operators for any $p\ge1$
