Pointwise complexity of the derivative of a computable function.

We explore the relationship between analytic behavior of a computable real valued function and the computability-theoretic complexity of the individual values of its derivative (the function's slopes) almost-everywhere. Given a computable function f, the values of its derivative f ′ (x) , where they are defined, are uniformly computable from x ′ , the Turing jump of the input. It is known that when f is C 2 , the values of f ′ (x) are actually computable from x. We construct a C 1 function f so that, almost everywhere, f ′ (x) ≥ T x ′ . Although the values f ′ (x) at each point x cannot uniformly compute the corresponding jumps x ′ of the inputs x almost everywhere for any C 1 function f, we produce an example of a C 1 function f such that f (x) ≥ T ∅ ′ uniformly on subsets of arbitrarily large measure, effectively (using the notion of a Schnorr test). We also explore analogous questions for weaker smoothness conditions, such as for f differentiable everywhere, and f differentiable almost everywhere. [ABSTRACT FROM AUTHOR]