A lightface analysis of the differentiability rank

Abstract

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than omega_1 which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal alpha>0, the set of Turing indices of C[0,1] functions that are differentiable with rank at most alpha is Pi_{2 alpha + 1}-complete. This result is expressed in the notation of Ash and Knight.