Page 254 - DMTH401_REAL ANALYSIS

Real Analysis

Notes b
(ii) L(P,f) <  f(x) dx - Î
a
for every partition P of [a,b] with P < d .

Proof: We consider (i). As f is bounded, there exists a positive number k such that
b
Î
f(x) £ k  x [a,b]. As f(x) dx is the infimum of the set of upper sums, therefore to each Î > 0,

a
there is a partition P of [a,b] such that
1
b Î
U(P ,f) <  f(x) dx + , (1)
1
a 2
Let P = {x , x , ...., x } and d be a positive number such that 2 k (p – 1) d = Î/2. Let P be a partition
1 0 1 p
.
of [a,b] with P < d Consider the common refinement P = P U P of P and P .
2 1 1
Each partition has the same end points 'a' and 'b'. So P is a refinement of P having at the most
2
(p – 1) more points than P. Consequently, by Theorem 3,
U(P,f) –2(p – 1) k d £ U(P ,f)
2
£ U(P ,f)
2
b
<  f(x) dx+ Î /2. (using (1))
a
Thus

b Î
U(P,f) <  f(x) dx + + 2(p 1) k d
-
a 2
6
=  f(x) dx+ Î , with P < d .
a

Task Write down the proof of part (ii) of Darboux's Theorem.

As mentioned earlier, Darboux's Theorem immediately leads us to the conditions of integrability.
We discuss this in the form of the following theorem:
Theorem 5: Condition of Integrability

First Form: The necessary and sufficient condition for a bounded function f to be integrable over
[a,b] is that to every number Î > 0 there corresponds d > 0 such that

-
U(P,f) L(P,f) < Î  P with P < d .
,
Proof: We firstly prove the necessity of the condition.
Since the bounded function f is integrable on [a, b], we have

b b b
 f(x) dx =  f(x) dx =  f(x) dx.
a a a
Let Î > 0 be any number. By Darboux Theorem, there is a number d > 0 such that

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