On any subinterval in the partition there a rational number, , and an irrational, .
So and giving the upper and lower values. (There are no other values.)
Does that help you see the answer?
Okay, so clearly (since ), so we can concentrate on .Define if is rational and if is irrational. Compute and .
The problem is, I am having trouble interpreting my textbook on how to calculate this. According to it,
: a partition of .
Okay, that's all very well. But how in the world do we compute that? Thus far, I have not been able to find a way.
Any help would be much appreciated. Thanks!
I will elaborate, there is a standard theorem which is not given is most basic developments of the Riemann Integral: if the set of discontinuities is not countable then the function is not integrable.
What is the set of discontinuities for this function?