  The paper studies the problem, for which continuous functions $f$ on the real
line ${\Bbb R}$, the difference of the functions $f(B)-f(A)$ of self-adjoint
operators $A$ and $B$ with trace class difference must also be of trace class.
The main result of the paper shows that this happens if and only if the
function $f$ is operator Lipschitz on a neighbourhood of zero.
