Ultrastability of nth minimal errors

We use the ultraproduct technique to study local properties of basic quantities of information-based complexity theory-the nth minimal errors. We consider linear and nonlinear operators in normed spaces; information consists of continuous linear functionals and is assumed to be adaptive. We establish ultrastability and disprove regularity of nth minimal errors. As a consequence, we answer a question posed by Hinrichs et al. in a recent paper [A. Hinrichs, E. Novak, H. Woźniakowski, Discontinuous information in the worst case and randomized settings, Math. Nachr. http://dx.doi:10.1002/mana.201100128].