Abstract We prove that, in a weak sense, any integrable function on the real line tends to zero at infinity: if f is an integrable function on ¡, then for almost all real number x, the sequence (f (nx ...

We present an example, different from Volterra's, of a bounded derivative that is not Riemann integrable. The existence of such functions was one of the motivations for Lebesgue to devise a stronger ...