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Recovering a Function from a Dini Derivative

provides a clear answer if we can assume that F' is Riemann integrable. Students of analysis will learn that if F' is Lebesgue integrable the same formula can be used, interpreting the integral in this more general sense. A full resolution of the problem requires a more general integral still, that of Denjoy and Perron (known frequently now as the Henstock-Kurzweil integral). The main question of this paper is, as it was for Lebesgue, whether a function can be recovered as an indefinite integral of one of its Dini derivatives?that is, when does the formula