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Pointwise complexity of the derivative of a computable function

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Abstract

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 \({\mathcal {C}}^2\), the values of \(f'(x)\) are actually computable from x. We construct a \({\mathcal {C}}^1\) function f so that, almost everywhere, \(f'(x)\ge _T x'\). Although the values \(f'(x)\) at each point x cannot uniformly compute the corresponding jumps \(x'\) of the inputs x almost everywhere for any \({\mathcal {C}}^1\) function f, we produce an example of a \({\mathcal {C}}^1\) function f such that \(f(x)\ge _T \emptyset '\) 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.

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Notes

  1. Depending on what convention the reader prefers for assigning values at the discontinuities of step functions, this may be taken to hold for all x.

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The author wishes to thank the anonymous referee for his/her valuable input.

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Correspondence to Ethan McCarthy.

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McCarthy, E. Pointwise complexity of the derivative of a computable function . Arch. Math. Logic 60, 981–994 (2021). https://doi.org/10.1007/s00153-021-00769-4

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