Page 255 - DMTH401_REAL ANALYSIS

Unit 20: The Riemann Integration

b b Notes
U(P,f) <  f(x) dx + Î /2 =  f(x)dx+ Î /2  P with P < d (2)
a a
Also,
b b
L(P,f) >  f(x) dx- Î /2 =  f(x) dx - Î /2 (3)
a a
b
i.e. L(P,f)- <  f(x) dx + Î /2  P with P < d
- a
Adding (2) and (3), we get

-
U(P,f) L(P,f) < Î P with P < d .
Next, we prove that condition is sufficient.
It is given that, for each number Î > 0, there is a number d > 0 such that

U(P,f) L(P,f) < Î  P with P < d .
,
-
.
Let P be a fixed partition with P < d Then
b b
L(P,f) £  f(x)dx £  f(x)dx £ U(P,f).
a a
b b
-
Therefore, f(x) dx -  f(x) dx £ U(P,f) L(P,f) <Î .

a a
Since Î is arbitrary, therefore the non-negative number
b b
 f(x) dx -  f(x) dx
a a
b b
is less than every positive number. Hence it must be equal to zero that is f(x) dx =  f(x) dx and

a a
consequently f is integrable over [a,b].
Second 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 a partition P of [a,b] such that

U (P,f) – L(P,f) < Î.

20.2 Riemann Integrable Functions

As we derived the necessary and sufficient conditions for the integrability of a function, we can
now decide whether a function is Riemann integrable without finding the upper and lower
integrals of the function. By using the sufficient part of the conditions, we test the integrability
of the functions. Here we discuss functions which are always integrable. We will show that a
continuous function is always Riemann integrable. The integrability is not affected even when
there are finites number of points of discontinuity or the set of points of discontinuity of the
function has a finite number of limit points. It will also be shown that a monotonic function is
also always Riemann integrable.

We shall denote by R(a,b), the family of all Riemann integrable functions on [a,b]. First we
discuss results pertaining to continuous functions in the form of the following theorems.

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