  In statistical physics lately a specific kind of average, called the
q-expectation value, has been extensively used in the context of q-generalized
statistics dealing with distributions following power-laws. In this context
q-expectation values appear naturally. After it has been recently shown that
this non-linear functional is instable, under a very strong notion of
stability, it is therefore of high interest to know sufficient conditions for
when the results of q-expectations are robust under small variations of the
underlying distribution function. We show that reasonable restrictions on the
domain of admissible probability distributions restore uniform continuity for
the q-expectation. Bounds on the size of admissible variations can be given.
The practical usefulness of the theorems for estimating the robustness of the
q-expectation value with respect to small variations is discussed.
