Example 10.2.1.
Define
Proof: We start with integrability of Consider the partition of where the partition in the direction is and in the direction The corresponding subrectangles are
and
The upper and lower sums are arbitrarily close and the lower sum is always zero, so the function is integrable and
For every fixed the function that takes to is zero except perhaps at a single point Such a function is integrable and Therefore, However, if the function that takes to is the nonintegrable function that is 1 on the rationals and 0 on the irrationals. See Example 5.1.4.