On the interpolation of discontinuous functions

Given a sequence of real numbers, we consider its subsequences converging to possibly different limits and associate to each of them an index of convergence which depends on the density of the associated subsequences. This index turns out to be useful for a detailed description of some phenomena in interpolation theory at points of discontinuity of the first kind. In particular we give some applications to Lagrange operators on Chebyshev nodes of the first and the second kind and Shepard operators.