The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix E of an elliptically symmetric t distribution on R d with degrees of freedom v > I extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U. Here U consists of P putting not too much mass in hyperplanes of dimension < d, as shown for empirical measures by Kent and Tyler [Ann. Statist. 19 (1991) 2102-2119]. It will be seen here that (μ, E) is analytic on U for the bounded Lipschitz norm, or for d = 1 for the sup norm on distribution functions. For k = 1, 2,..., and other norms, depending on k and more directly adapted to t functionals, one has continuous differentiability of order k, allowing the delta-method to be applied to (μ, Σ) for any P in U, which can be arbitrarily heavy-tailed. These results imply asymptotic normality of the corresponding M-estimators (μ n , Σ n In dimension d = 1 only, the t v functional (μ, a) extends to be defined and weakly continuous at all P.