The Riemann IntegralMATH 464/506, Real Analysis

J. Robert Buchanan

Department of Mathematics

Summer 2007

J. Robert Buchanan The Riemann Integral

Partitions

Definition

A partition of an interval I = [a, b] is a collectionP = {I1, . . . , In} of non-overlapping closed intervals whoseunion is [a, b]. We ordinarily denote the intervals byIi = [xi−1, xi ], where

a = x0 < · · · < xi−1 < xi < · · · < xn = b.

The points xi (i = 0, 1, . . . , n) are called the partition points ofP. The norm or mesh of P is the number

‖P‖ = max{x1 − x0, x2 − x1, . . . , xn − xn−1}.

J. Robert Buchanan The Riemann Integral

Tagged Partitions

Definition

If a point ti has been chosen from each interval Ii , fori = 1, . . . , n, then the points ti are called the tags and the set ofordered pairs

P = {(I1, t1), . . . , (In, tn)}is called a tagged partition of I.

J. Robert Buchanan The Riemann Integral

Riemann Sums

Definition

If P is a tagged partition of [a, b], the Riemann sum of afunction f : [a, b] → R corresponding to P is the number

S(f ; P) =n

∑

i=1

f (ti)(xi − xi−1).

J. Robert Buchanan The Riemann Integral

Riemann Integral

Definition

A function f : [a, b] → R is said to be Riemann integrable on[a, b] if there exists a number L ∈ R such that for every ǫ > 0there exists δǫ > 0 such that if P is any tagged partition of [a, b]with ‖P‖ < δǫ, then

|S(f ; P) − L| < ǫ.

The set of all Riemann integrable functions on [a, b] will bedenoted by R[a, b].

Notation:

L =

∫ b

af =

∫ b

af (x) dx

J. Robert Buchanan The Riemann Integral

Uniqueness of the Riemann Integral

Theorem

If f ∈ R[a, b], then the value of the integral is uniquelydetermined.

Proof.

J. Robert Buchanan The Riemann Integral

Uniqueness of the Riemann Integral

Theorem

If f ∈ R[a, b], then the value of the integral is uniquelydetermined.

Proof.

J. Robert Buchanan The Riemann Integral

Examples

Example

1 f (x) = k ∈ R[a, b]

2 g(x) =

{

2 if 0 ≤ x ≤ 2,1 if 2 < x ≤ 3.

3 h(x) = x ∈ R[0, 1]

4 j(x) =

{

1/n if x = 1/n where n ∈ N,0 otherwise.

J. Robert Buchanan The Riemann Integral

Properties of the Riemann Integral

Theorem

Suppose that f and g are in R[a, b]. Then:1 If k ∈ R, the function kf ∈ R[a, b] and

∫ b

akf = k

∫ b

af .

2 The function f + g is in R[a, b] and

∫ b

a(f + g) =

∫ b

af +

∫ b

ag.

3 If f (x) ≤ g(x) for all x ∈ [a, b], then

∫ b

af ≤

∫ b

ag.

Proof. J. Robert Buchanan The Riemann Integral

Properties of the Riemann Integral

Theorem

Suppose that f and g are in R[a, b]. Then:1 If k ∈ R, the function kf ∈ R[a, b] and

∫ b

akf = k

∫ b

af .

2 The function f + g is in R[a, b] and

∫ b

a(f + g) =

∫ b

af +

∫ b

ag.

3 If f (x) ≤ g(x) for all x ∈ [a, b], then

∫ b

af ≤

∫ b

ag.

Proof. J. Robert Buchanan The Riemann Integral

Boundedness and Integrability

Theorem

If f ∈ R[a, b], then f is bounded on [a, b].

Proof.

J. Robert Buchanan The Riemann Integral

Boundedness and Integrability

Theorem

If f ∈ R[a, b], then f is bounded on [a, b].

Proof.

J. Robert Buchanan The Riemann Integral

Thomae’s Function Revisited

Example

Suppose

f (x) =

{

1/n if x ∈ Q ∩ (0,∞), x = mn , and gcd(m, n) = 1

0 if x ∈ R\Q.

f (1) = 1

f (√

2) = 0

f (3/2) = 1/2

J. Robert Buchanan The Riemann Integral

Thomae’s Function (cont.)

���

1

21 ���

3

22

1

���

1

2

���

1

3

���

1

4���

1

5

Thomae’s function is continuous at every positive irrationalnumber and discontinuous at every positive rational number,but it is Riemann integrable.

J. Robert Buchanan The Riemann Integral

Homework

Read Section 7.1

Page 201-202: 1, 2, 5, 8, 11, 16

J. Robert Buchanan The Riemann Integral