In this paper we derive the change-of-variance function of M-estimators of scale under general contamination, thereby extending the formula in Hampel et al. (1986). We say that an M-estimator is B-robust if its influence function is bounded, and we call it V-robust if its change-of-variance function is bounded from above. It is shown, for a natural class of M-estimators, that the general notion of V-robustness still implies B-robustness. Several classes of M-estimators are studied closely, as well as some typical examples and their interpretation.