Question

prove that f(x) = [x] is integrable on [0,3]​

Answer 1

Answer:

I think that the best thing to do is prove that the upper and lower sums are equal in the limit. Since f is monotonic I know that for any partition {x0,…,xN} the upper and lower sums are given by

U=∑i=1Nxi(xi−xi−1)

and

L=∑i=1Nxi−1(xi−xi−1)

respectively. I considered showing that the the limit of U−L as N→∞ is 0, hoping that I would get some kind of telescoping situation, but that doesn't seem to be happening:

U−L=∑i=1N(xi−xi−1)(xi−xi−1)

I can't see a nice way to show that that is going to be less than any ϵ. Does this seem like the right approach? Am I missing something?