The Complexity of Antidifferentiation, Denjoy Totalization, and Hyperarithmetic Reals

We consider real functions on the interval [0, 1]. Denote by Δ the set of derivatives; i.e., Δ = {ƒ:ƒ is a derivative} = {ƒ : ∃F: [0,1]→ R {F is differentiable and ƒ = F')}. If ƒ є Δ, any F with F' = ƒ is a primitive of ƒ and is uniquely determined up to a constant. To normalize, we denote by F(x) = ʃ^x)0 ƒ the unique primitive of ƒ with F(0) = 0. This is the original Newtonian concept of integration as the inverse operation of differentiation, i.e., antidifferentiation.