A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK

Abstract

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 α is ${{\rm{\Pi }}_{2\alpha + 1}}$ -complete. This result is expressed in the notation of Ash and Knight.