Define if is rational and if is irrational.
Is Riemann Integrable on ?
the lower sum should be zero since in any subdivision, there is an irrational number. and if each subdivision is so small, then upper sum could be really small. so i think the integral is 0
but if you do the samething to Dirichlets funtion, shouldnt the integral be 0 also? but Dirichlets funtion is not Riemann integrable.
i dont see the difference in these functions.
it is not continuous on (0,1], so it is not Riemann integrable.
There is a theorem which finally solve the problem to judge which limitary function is integrable: the set of uncontinuous points forms a "null set",that is, we can use countable numbers of open interval to cover a real numbers' set completely and the sum of the "length" of these interval ( )can be arbitrarily close to 0.
A obvious corollary is that if the uncontinuous points set forms an interval (a,b), then the function must not be Riemann integrable.
for every irrational number , we can use a sequence of rational numbers to approach it. In these sequence, the denominator would approach to . So the result will have limit 0. So it will be continuous at . So the uncontinous points set would be countable.
Let we list these points by , then the sum of the length of the sequence would be , which will have limit 0 when . So it would be a "null set". So the function is Riemann integrable