Prove

* Proof. * From this exercise (1.11 #4, part b) we know

But, since is constant on the open subintervals of the partition

which contains every integer between and , we have on the open subintervals of (since there are no integers in the open subintervals). Hence, . Thus,

open subintervals of P does not contain integers, I agree. But why did you ignore integer points in [a,b] and concluded that [-x] = -[x]-1 always ?

The partition does not contain all integers. For example, let a = 1.9 and b =3.8. [a] = 1, b-a = 1.9, and [b-a] = 1. Therefore, your proposed partition is 1.9, 2, 3.8. This is missing the integer 3.

The partition does contain all integers including the integers in the edges of the interval, its the open subintervals of the Partition which define the subdivision points of the integration procedure, those are inmaterial.