<p>Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood examples in this sense. We analyze the continuity of global solutions as functions of the finite interpolation data in neighbourhoods of elements distinguished by this uniqueness property. Our study covers rational inner or Cayley rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.</p>