Starting from a real-data motivation, we explore continuity of methods based on halfspace depth - as related to numerical stability and robustness. We consider the multivariate location setting, as well as the more recent regression depth. For depth contours (which we understand as upper level sets) and maximum depth estimators, we investigate separately possibility of implosion and explosion. We obtain not only theorems concerning "typical" cases, but also some algorithmically verifiable conditions. Based on these, some proposals concerning use and possible diagnostics of depth-related methods are given.