The set of asymptotic values of a continuous function on the open unit disc in ℝ2 forms an analytic set, in the sense of being a continuous image of a Polish space (complete, separable metric space). This was proved in [9] by J. E. McMillan, who had earlier given versions of this result for holomorphic and meromorphic functions. We extend his method to the case of a function on the open unit ball of ℝn which is continuous merely in the fine topology, the coarsest topology making all subharmonic functions continuous. In particular, we use a version of McMillan's ingenious metric on a certain space of equivalence classes of asymptotic paths. McMillan also proved in [9] that the set of point asymptotic values of a continuous function in the unit disc forms an analytic set. We use a modification of the McMillan metric to extend this result to fine continuous functions in the unit ball and deduce that the set of boundary points of the unit ball at which the function has an asymptotic value forms an analytic set.