Page 248 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Let f be a real function defined and bounded on a closed interval [a,b].
Recall that a real function f is said to be bounded if the range of f is a bounded subset of R, that
is, if there exist numbers m and M such that m £ f(x) £ M for each x Î[a,b]. M is an upper bound
and m is a lower bound of f in [a,b]. You also know that when f is bounded, its supremum and
infimum exist. We introduce the concept of a partition of [a,b] and other related definitions:

Definition 1: Partition
Let [a,b] be a given interval. By a partition P of [a,b] we mean a finite set of points {x , x , ...., x,},
0 l
where
a = x , < x <… < x <x = b.
0 1 n-1 n
We write x = x – x , (i=l, 2, ..., n). So x is the length of the ith sub-interval given by the
i i i-1 i
partition P.
Definition 2: Norm of a Partition
Norm of a partition P, denoted by P , is defined by P = max Ax,. Namely, the norm of P is the
t £ £ n
i
length of largest sub-interval of [a, b] induced by P. Norm of P is also denoted by (P).
There is a one-to-one correspondence between the partitions of [a,b] and finite subsets of ]a, b[.
This induces a partial ordering on the set of partitions of [a,b]. So, we have the following
definition.
Definltion 3: Refinement of a Partition
Let P, and P, be two partitions of [a,b]. We say that P, is finer than P, or P , refines P, or P is a
2 2
refinement of P if P  P , that is, every point of P is a point of P,.
1 1 2 1
You may note that, if P, and P , are any two partitions of [a,b], then P,  P is a common
2 2
}
}
1 1 1
1 1 1 1 1
refinement of P, and P . For example, if P = { 0, , , ,1 and P = { 0, , , , , ,1 are
2 1 4 3 2 2 6 5 4 3 2
}
1 1 1 1 1
partitions of [0, 1], then P is a refinement of P and P  P = { 0, , , , , ,1 is their common
2 1 1 2 6 5 4 3 2
refinement.
We now introduce the notions of upper sums and lower sums of a bounded function f on an
interval [a, b], as given by Darboux. These are sometimes referred to as Darboux Sums.
Definition 4: Upper and Lower Sums
Let f: [a,b]  R be a bounded function, and let P = (x , X ... x,) be a partition of [a,b]. For i = 1, 2
0 1
....., n, let M and m be defined by
i i
M = lub (f(x) : x £x £ x )
i i-1 i
m = glb (f(x) : x £ x £ x )
i i-1 i
i.e. M and m be the supremum and infimum of f in the sub-interval [x , x ].
i i i-1 i
Then, the upper (Riemann) sum of f corresponding to the partition P, denoted by U (P,f), is
defined by
n
U(P,f) = å M  x i
i
=
i 1

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