"Can the author prove that case (3): one is less than R and one is greater than R occurs with greater than zero probability?" I think it falls out of the way y

"Can the author prove that case (3): one is less than R and one is greater than R occurs with greater than zero probability?" I think it falls out of the way you pick R. If you pick from a distribution with a non-zero probability everywhere, then for any specific (non-equal) X and Y, you've got non-zero probability integrating between them.