Dirichlet Function
Equal to at rationals and at irrationals; every subinterval contains both, so every upper sum is and every lower sum is , and the function is not Riemann integrable.
Example 4.2
Consider, on , the function
No matter what we take, we always have rationals and irrationals in each subinterval , hence
Failure of integrability
On the upper and lower integrals of the Dirichlet function are and , so is not Riemann integrable.
This contrasts with the Thomae function, which is also nonzero at infinitely many, indeed dense many points, yet is integrable: there the values at the bad points tend to , so the upper sums can be made arbitrarily small.
Related
Stated in
- Example 4.2§4.1 Basics