ABSTRACT
A refinement of P is a partition containing P. The common refinement of partitions P and Q is the partition P ∪Q. ♦ 5.1.2 Example. Let p ∈ N. Then, for each n ∈ N,
Pn := {j/pn : j = 0, 1, . . . , pn} is a partition of [0, 1], ‖Pn‖ = p−n, and Pn+1 is a refinement of Pn. ♦ 5.1.3 Definition. The lower and upper (Darboux) sums of f over a partition P of [a, b] are defined, respectively, by
S(f,P) := n∑ j=1
mj∆xj and S(f,P) := n∑ j=1
Mj∆xj ,
where
mj = mj(f) := inf xj−1≤x≤xj
f(x) and Mj = Mj(f) := sup xj−1≤x≤xj
f(x). ♦
A geometric interpretation of the upper and lower sums for a positive continuous function is given in Figure 5.1. The lower (upper) sum is the total area of the smaller (larger) rectangles.
