Functions not first-return integrable ✩

Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0, 1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C∞ on (0, 1] yet fails to be first-return integrable