A Lightface Analysis of the differentiability rank

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 ω1 which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal α > 0, the set of Turing indices of C[0, 1] functions that are differentiable with rank at most α is Π2α+1-complete. This result is expressed in the notation of Ash and Knight.