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.)
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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