Ultrastability of nth minimal errors

We use the ultraproduct technique to study local properties of basic quantities of information-based complexity theory – the n-th 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 n-th minimal errors. As a consequence, we answer a question posed by Hinrichs, Novak, and Woźniakowski in a recent paper (Discontinuous information in the worst case and randomized settings, Math. Nachr., doi:10.1002/mana.201100128).