A new proof of Fleissner's theorem on products of derivatives (Q1096027)

Let I be an interval of the real line. Saying that \(f:I\to {\mathbb{R}}\) is a derivative means there exists a differentiable function \(F:I\to {\mathbb{R}}\) such that \(F'(x)=f(x),\quad x\in I.\) In this case, F will be called a primitive of f. Let \(\Delta =\{f:I\to {\mathbb{R}}|\) f is a derivative\(\}\) and \(B=\{g:I\to {\mathbb{R}}| f\cdot g\in \Delta\) for every \(f\in \Delta \}\). \textit{R. J. Fleissner} [Fundam. Math. 88, 173-178 (1975; Zbl 0308.26006)] using the theory of Denjoy integral showed that every continuous function which is of bounded variation belongs to B. In the present paper we give a simple proof of Fleissner's result, which does not involve the theory of the Denjoy integral.