The characteristic function of the RATIONALS is a well-known example of a bounded function that is not Riemann integrable. But is the characteristic function of the IRRATIONALS (that is, the function that is 1 at every irrational number and 0 at every rational number) Riemann integrable on an arbitrary interval [a,b]? It seems like it would be, and that its integral would be equal to 1. But maybe I'm wrong. Anyone know for a fact?