Integrability conditions for approximate derivatives (Q1073935)

Let F be a real function on [0,1]. Let \(F^*\) equal F' wherever it exists and 0 elsewhere. Let F be approximately differentiable. Then \(F^*\) is Denjoy integrable in the restricted sense if and only if \(F'_{ap}\) is integrable in the same sense. Two counterexamples show that the wide Denjoy integral cannot replace the restricted one in the above theorem.