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.