A remark on absolutely continuous functions (Q919117)

A Lusin type theorem for absolutely continuous functions was obtained by the authors in a previous paper [same journal 5, 261-266 (1980; Zbl 0474.26005)]. The theorem states that f on [0,1] is absolutely continuous if and only if for every \(\epsilon >0\) there is a continuously differentiable function g such that the set E for which \(f(x)\neq g(x)\) has measure less than \(\epsilon\) and \(\int_{E}| f'(t)| dt<\epsilon,\int_{E}| g'(t)| dt<\epsilon.\) In this paper it is shown that the result is in a certain sense the best possible.