The Right-Hand Derivative of an Integral

In this note, we provide a general result that explains the examples contained in [1], [3], [4], and [5]. We also observe with interest that this problem seems to have been proposed anew at nearly regular intervals of ten to fifteen years. The article [2] also contains a discussion of this problem. Using the notation in [3], suppose that f is a real-valued function defined on the interval [0, 1] such that (a) f is bounded on [0, 1], (b) f is continuous on (0, 1], and (c) limx→0+ f (x) does not exist. Then let F(x) = ∫ x 0 f (t) dt for x in [0, 1]. The question is whether or not the right-hand derivative