Page 256 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Theorem 6: If f: [a, bJ  R is a continuous function, then f is integrable over [a, b], that is
f Î R(a,b).
Proof: If f is a continuous function on [a,b] then f is bounded and is also uniformly continuous.

To show that f Î R [a,b] you have to show that to each number Î > 0, there is a partition P for
which
U(P,f) – L(P,f) < Î

Let Î> 0 be given. Since f is uniformly continuous on [a,b], there is a number d > 0 such that
E
-
f(x) f(y) < whenever x y < d Let P be any partition of [a,b] with P < d .
.
-
b a
-
We show that, for such a partition P, U (P, f) – L(P, f) < Î.
n n
Now, U(P,f) – L(P,f) = å M  x - å m  x i
i
i
i
i 1 i 1
=
=
n

= å (M - m ) x , (4)
1
i
i
i 1
=
x
where x i = x - x i 1 and M = sup {f(x) x i 1 £ x £ x 1 } = f( ) 1 [x i 1 ,x ]. Such
,
1 (say), for same x Î
-
-
1
-
i
i
a x exists because a continuous function f attains its bounds on [x – x ].
i i-1 1
Similarly, m = inf {f(x) x £ x £ x } = f( ) (say), for some h Î [x ,x ]. Hence
h
-
-
i i 1 i i i i 1 i
-
h
-
x
M - m = f( ) f( ) £ f( ) f( ) < Î /b a, for all i,
-
x
h
i
i
i
i
i
i
.
since x - h £ A x < d Substituting in (4) we obtain
i
i
i
n

U(P,f) – L(P,f) = å (M - m ) x i
i
i
=
i 1
Î n
< ( å  x i )
b a i 1
-
=
E
-
(b a) = Î .
b a
-
Thus, every continuous function is Riemann integrable,
But as remarked earlier, even when there are discontinuous of the function, it is integrable. This
is given in the next two concepts which we state without proof.
Theorem 7: Let the bounded function f: [a, b]  R have a finite number of discontinuities. Then
f Î R (a,b).
Theorem 8: Let the sec of points of discontinuity of a, bounded function f: [a, b]  R has a finite
number of limit points, then f Î R (a, b).
We illustrate these theorems with the help of examples.
2
Example: Show that the function f where f(x) = x is integrable in every interval [a,b].
2
Solution: You know that the function f(x) = x is continuous. Therefore it is integrable in every
interval [a,b].
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