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.

我们探讨了可计算的实值函数的分析行为与其几乎所有位置的导数（函数的斜率）的各个值的可计算性-理论复杂性之间的关系。给定一个可计算的函数f，可以根据输入的图灵跳转\（x'\）统一计算其导数\（f'（x）\）的值（在此处定义它们）。众所周知，当f为\（{\ mathcal {C}} ^ 2 \）时，\（f'（x）\）的值实际上可以从x计算得到。我们构造了一个\（{\ mathcal {C}} ^ 1 \）函数f，以便几乎在任何地方，\（f'（x）\ ge _T x'\）。尽管每个点x的值\（f'（x'\）不能均匀地为任何\（{\ mathcal {C}} ^ 1 \）函数来计算输入x的相应跳跃\（x'\）f，我们生成一个\（{\ mathcal {C}} ^ 1 \）函数f的示例，使得\（f（x）\ ge _T \ emptyset'\）均匀地分布在任意大尺度子集上，有效地（使用Schnorr测试的概念）。我们还针对较弱的光滑度条件（例如针对f随处可微）和f 几乎无处不在。