CHAPTER 5 INTEGRABLITY ON R
5.1 RIEMANN INTEGRAL
DEFINITION. Let a; b 2 R with a < b.
(1) A partition of the interval [a; b] is a set of points P = fx0 ; x1 ; ¢ ¢ ¢ ; xn g such
that
a = x0 < x1 < ¢ ¢ ¢ < xn = b:
(2) The norm of a partition P = fx0 ; x1 ; ¢ ¢ ¢ ; xn g is the number
kP k = max jxj ¡ xj¡1 j:
1·j·n
(3) A re¯nement of a partition P is a partition Q of [a; b] that satis¯es P µ Q.
In this case we say that Q is ¯ner than P .
Example. [Dyadic Partition] Pn = f 2jn : j = 0; 1; ¢ ¢ ¢ ; 2n g is a partition on [0; 1],
and Pm is ¯ner than Pn if m ¸ n.
Remark. If P and Q are partitions of [a; b], the P [ Q is ¯ner than both P and
Q. If Q is a re¯nement of P then kQk · kP k:
DEFINITION. Let a; b 2 R with a < b and let P = fx0 ; x1 ; ¢ ¢ ¢ ; xn g be a partition
of the interval [a; b], and suppose that f : [a; b] ! R is bounded.
(1) The upper Riemann sum of f over P is the number
U (f; P ) =
n
X
j=1
Mj (xj ¡ xj¡1 );
where
Mj =
sup
f (x):
x2[xj¡1 ;xj ]
(2) The lower Riemann sum of f over P is the number
L(f; P ) =
n
X
j=1
mj (xj ¡ xj¡1 );
where
mj =
inf
x2[xj¡1 ;xj ]
f (x):
Remark. If g : N ! R, and m > n, then
m
X
(g(k + 1) ¡ g(k)) = g(m + 1) ¡ g(n):
k=n
Typeset by AMS-TEX
1
2
Remark. If f (x) = c is constant on [a; b] then
U(f; P ) = L(f; P ) = c(b ¡ a);
for all partition P of [a; b].
Remark. L(f; P ) · U (f; P ) for all partitions P and all bounded functions f .
Remark. If P is a partition of [a; b] and Q is a re¯nement of P , then
L(f; P ) · L(f; Q) · U (f; Q) · U (f; P ):
Remark. If P and Q are partitions of [a; b] , then L(f; P ) · U (f; Q):
DEFINITION. let a; b 2 R with a < b. A function f : [a; b] ! R is said to be
(Riemann) integrable on [a; b] if and only if f is bounded on [a; b] and for any
² > 0 there is a partition P of [a; b] such that U (f; P ) ¡ L(f; P ) < ²:
Theorem. Suppose that a; b 2 R with a < b. If f is continuous on the interval [a; b], then
f is integrable on [a; b].
Example. The Dirichlet function.
Example. f (x) = 0 for 0 · x <
1
2
and f (x) = 1 for
1
2
· x · 1.
DEFINITION. let a; b 2 R with a < b and f : [a; b] ! R be bounded.
(1) The upper integral of f on [a; b] is the number
Z b
f (x)dx = inffU (f; P )g
(U )
a
where P is partition of [a; b].
(2) The lower integral of f on [a; b] is the number
Z b
(L)
f (x)dx = supfL(f; P )g
a
where P is partition of [a; b].
(3) If the upper and lower integrals of f on [a; b] are equal, we de¯ne the
integral of f on [a; b] to be the common value
Z b
Z b
Z b
f (x)dx = (U )
f (x)dx = (L)
f (x)dx:
a
a
a
Remark. If f : [a; b] ! R is bounded, then the upper and lower integrals exist
and are ¯nite, and satisfy
Z b
Z b
(L)
f (x)dx · (U )
f (x)dx:
a
a
3
Theorem. let a; b 2 R with a < b and f : [a; b] ! R is bounded. Then f is integrable
on[a; b] if and only if
Z b
Z b
(L)
f (x)dx = (U )
f (x)dx:
a
a
Theorem. If f (x) = c on [a; b] is a constant, then
Z
b
a
f (x)dx = c(b ¡ a):
5.2 RIEMANN SUM
DEFINITION. Let f : [a; b] ! R.
(1) A Riemann sum of f with respect to a partition P = fx0 ; x1 ; ¢ ¢ ¢ ; xn g of
[a; b] is a sum of the form
n
X
j=1
f (tj )(xj ¡ xj¡1 );
where the choice of tj 2 [xj¡1 ; xj ] is artitrary.
(2) The Riemann sum is said to converge to I(f ) as kP k ! 0 if and only if
given ² > 0 there is partition P² of [a; b] such that P = fx0 ; ¢ ¢ ¢ ; xn g ¾ P²
implies
n
X
j
f (tj )(xx ¡ xj¡1 ) ¡ I(f )j < ²
j=1
for all choices tj 2 [xj¡1 ; xj ]; j = 1; 2; ¢ ¢ ¢ ; n. In this case we shall use the
notation
n
X
I(f ) = lim
f (tj )(xj ¡ xj¡1 ):
kP k!0
j=1
Theorem. let a; b 2 R with a < b and f : [a; b] ! R is bounded. Then f is Riemann
integrable on [a; b] if and only if
I(f ) = lim
kP k!0
n
X
j=1
exists, in which case
I(f ) =
Z
f (tj )(xj ¡ xj¡1 )
b
f (x)dx:
a
4
Theorem. [Linear property] If f; g are integrable on [a; b] and c 2 R then f + g and
cf are integrable on [a; b]. In fact
Z
Z
b
(f + g)(x)dx =
a
b
f (x)dx +
a
and
Z
Z
b
g(x)dx
a
b
cf (x)dx = c
a
Z
b
f (x)dx:
a
Theorem. If f is integrable on [a; b] then f is integrable on each subinterval [c; d] of [a; b].
Moreover
Z
Z
Z
b
c
f (x)dx =
b
f (x)dx +
a
a
f (x)dx
c
for all c 2 (a; b).
Remark. De¯ne
Rb
a
Ra
f (x)dx = ¡
b
Ra
f (x)dx and
a
f (x)dx = 0:
Theorem. [Comparison Theorem] If f; g are integrable on [a; b] and f (x) · g(x) for
all x 2 [a; b], then
Z b
Z b
f (x)dx ·
g(x)dx:
a
a
In particular. if m · f (x) · M for x 2 [a; b] then
m(b ¡ a) ·
Z
a
b
f (x)dx · M (b ¡ a):
Theorem. If f is integrable on [a; b], then jf j is integrable on [a; b] and
j
Z
b
a
f (x)dxj ·
Z
b
jf (x)jdx:
a
Corollary. If f and g are integrable on [a; b] then so is f g.
Theorem. [First Mean Value Theorem] Suppose f and g are integrable on [a; b] with
g(x) ¸ 0 for all x 2 [a; b]. If
m = inf f (x); M = sup f (x);
x2[a;b]
then there is a number c 2 [m; M ] such that
Z
a
b
f g(x)dx = c
x2[a;b]
Z
b
g(x)dx:
a
5
In particular if f is continuous on [a; b], then there is x0 2 [a; b] such that
Z
Z
b
f g(x)dx = f (x0 )
a
Example. Find F (x) =
b
g(x)dx:
a
Rx
f (t)dt where f (x) = 1 if x ¸ 0 and f (x) = ¡1 if x < 0.
Rx
Theorem. If f is integrable on [a; b], then F (x) = a f (t)dt exists and is continuous on
[a; b].
0
Theorem. [Second Mean Value Theorem] Suppose f and g are integrable on [a; b]
with g(x) ¸ 0 for all x 2 [a; b], and that m; M are real numbers that satisfy m ·
inf x2[a;b] f (x); M ¸ supx2[a;b] f (x). Then there exist an x0 2 [a; b] such that
Z
b
f g(x)dx = m
a
Z
x0
g(x)dx + M
a
Z
b
g(x)dx:
x0
In particular if f (x) ¸ 0 for all x 2 [a; b], then there is an x0 2 [a; b] that satis¯es
Z
b
f g(x)dx = M
a
Z
b
g(x)dx:
x0
5.3 FUNDAMENTAL THEOREM OF CALCULUS
Thoerem. Let [a; b] be nondegenerate and suppose that f : [a; b] ! R.
Rx
(1) If f is continuous on [a; b] and F (x) = a f (t)dt, then F 2 C 1 [a; b] and
F 0 (x) = f (x)
for each x 2 [a; b]:
(2) If f is di®erentiable on [a; b] and f 0 is integrable on [a; b], then
Z x
f 0 (t)dt = f (x) ¡ f (a)
a
for each x 2 [a; b]:
Remark.
(1) f (x) = ¡1 for x < 0 and f (x) = 1 for x ¸ 0:
(2) f (x) = x2 sin x12 on [0; 1]:
Example. (i)
R1
0
(3x ¡ 2)2 dx, (ii)
R
2
¼
0
(1 + sin x)dx:
6
Theorem. [Integration by Parts]. Suppose that f; g are di®erentiable on [a; b] with
f 0 ; g 0 are integrable on [a; b]. Then
Z b
Z b
0
f g(x)dx = f (b)g(b) ¡ f (a)g(a) ¡
f g 0 (x)dx:
a
Example.
Example.
R
¼
2
x sin xdx:
0
R3
1
a
log xdx:
Theorem. [Change of Variable]. Let Á be continuously di®erentiable on a nondegenerate closed interval [a; b]. If f is continuous on Á([a; b]) or if Á is strictly increasing on
[a; b] and f is integrable on [Á(a); Á(b)], then
Z Á(b)
Z b
f (t)dt =
f (Á(x))Á0 (x)dx:
Á(a)
Example.
Example.
R1
0
R1
p
1+x
e
¡1
a
p
= 1 + xdx:
xf (x2 )dx:
5.4 IMPROPER RIEMANN INTEGRATION
DEFINITION. Let (a; b) be a nonempty open (possibly unbounded) interval
and f : (a; b) ! R.
(1) f is said to be locally integrable on (a; b) if and only if f is integrable on
every closed subinterval [c; b] of (a; b).
(2) f is said to be improperly integrable on (a; b) if and only if f is locally
integrable on (a; b) and
Z d
Z b
f (x)dx = lim ( lim
f (x)dx)
a
c!a+ d!b¡
c
exists and ¯nite. The limit is called the improper integral of f on (a; b):
Remark. The order of the limit does not matter.
R1
Example. 0 p1x dx.
R1
Example. 1 x12 dx:
Theorem. If f; g are improperly integrable on (a; b) and ®; ¯ 2 R, then ®f + ¯g is also
improperly integrable on (a; b) and
Z b
Z b
Z b
(®f + ¯g)(x)dx = ®
f (x)dx + ¯
g(x)dx:
a
a
a
7
Theorem. [Comparison Theorem]. Suppose that f; g are locally integrable on (a; b),
and 0 · f (x) · g(x) for all x 2 (a; b). If g is improperly integrable on (a; b), then f is
improperly integrable on (a; b) and
Z
a
Example.
Example.
R1
0
b
f (x)dx ·
Z
b
g(x)dx:
a
j sin xj
p
:
x3
R1
1
log
p x dx:
x5
Remark. If f is bounded and locally integrable on (a; b) and jgj is improperly integrable
on (a; b), then jf gj is improperly integrable on (a; b).
DEFINITION. Let (a; b) be a nonempty open interval and f : (a; b) ! R.
(1) f is said to be absolutely intgrable on (a; b) if and only jf j is improperly
integrable on (a; b):
(2) f is said to be conditionally on (a; b) if and only if f is improperly integrable on (a; b), but not absolutely integrable on (a; b).
Theorem. If f is absolutely integrable on (a; b), then f is improperly integrable on (a; b)
and
Z b
Z b
f (x)dxj ·
jf (x)jdx:
j
a
Example.
R1
1
a
sin x
x dx:
5.5 FUNCTIONS OF BOUNDED VARIATION
DEFINITION.
Let f : [a; b] ! R. For any partition P = fx0 ; ¢ ¢ ¢ ; xn g of [a; b] let
Pn
V (f; P ) = 1 jf (xj ) ¡ f (xj¡1 )j.
V ar[a;b] (f ) := supfV (f; P ) : P is partition of [a; b]:g
f is said of bounded variation on [a; b] if and only if V ar[a;b] (f ) < 1:
Remark. If f is C 1 on [a; b], then f is of bounded variation on [a; b].
Example. x2 sin x1 :
Remark. If f is monotone on [a; b], then f is of bounded variation on [a; b].
Remark. If f is of bounded variation on [a; b], then f is bounded on [a; b].
Example. f (x) = sin x1 on (0; 1] and f (0) = 0.
8
Theorem. If f; g are of bounded variation on [a; b], then so are f + g; f ¡ g; cf .
DEFINITION. Let f be of bounded variation on [a; b], then the total variation
of f is the function F (x) = V ar[a;x] (f ):
Theorem. Let f be of bounded variation on [a; b] and let F be its total variation funvtion.
Then
(1) jf (y) ¡ f (x)j · F (y) ¡ F (x) for all a · x · y · b.
(2) F and F ¡ f are increasing on [a; b].
(3) V ar[a;b] (f ) · V ar[a;b] (F ):
Corollary. f is of bounded variation on [a; b] if and only if there are increasing functions
g; h on [a; b] such that f = g ¡ h
Remark. If f is of bounded variation on [a; b], then
(1) For all x 2 [a; b), f (x+) exists and for all x 2 (a; b], f (x¡) exists.
(2) f has at most countable may points of discontinuity.
(3) f is integrable on [a; b].
5.6 CONVEX FUNCTIONS
DEFINITION. Let I be an interval and f : I ! R.
(1) f is said to be convex on I if and only if
f (®x + (1 ¡ ®)y) · ®f (x) + (1 ¡ ®)f (y)
for all 0 · ® · 1 and all x; y 2 I:
(2) f is said to be concave on I if and only if ¡f is convex on I.
Remark. Let I be an interval and f : I ! R. Then f is convex on I if and only
if given any [c; d] ½ I, the chord through the points (c; f (c)); (d; f (d)) lies on or
above the graph y = f (x) for all x 2 [c; d]:
Example. jxj; x2
Remark. A function f is convex on a nonempty open interval (a; b) if and only
if the slope of the chord always increases on (a; b), a.e. a < c < x < d < b implies
f (x) ¡ f (c)
f (d) ¡ f (x)
·
:
x¡c
d¡x
Theorem. Suppose that f is di®erentiable on a nonempty open interval I. Then f is
convex on I if and only if f 0 is increasing on I.
9
Theorem. If f is convex on a nonempty, open interval I, then f is continuous on I.
Remark. f (x) = 0 if x 2 [0; 1) and f (1) = 1:
DEFINITION. f is said to have a proper maximum (respectively minimum)
at x0 if and only if there is a ± > 0 such that jx ¡ x0 j < ± implies f (x) < f (x0 )
(respectively f (x) > f (x0 ).
Theorem.
(1) If f is convex on a nonempty open interval (a; b), the f has no proper maximun on
(a; b).
(2) If f is convex on [0; 1) and has a proper minimum , then f (x) ! 1 as x ! 1:
Theorem. [Jensen's inequality]. Let Á be convex on a closed interval [a; b] and f :
[0; 1] ! [a; b]. If f and Á ± f are integrable on [0; 1], then
Z
Á(
1
0
f (x)dx) ·
Z
1
Á(f (x))dx:
0
DEFINITION. Let f : (a; b) ! R and x 2 (a; b).
(1) f is said to have a right-hand derivative at x if and only if
f (x + h) ¡ f (x)
h!0+
h
DR f (x) = lim
exists as an extended real number.
(2) f is said to have a left-hand derivative at x if and only if
f (x + h) ¡ f (x)
h!0¡
h
DL f (x) = lim
exists as an extended real number.
Remark. A real function f is di®erentiable at x if and only if both DR f (x); DL f (x)
exist , are ¯nite and equal, in which case f 0 (x) = DR f (x) = DL f (x):
Theorem. Let f be convex on an open interval (a; b) . Then the left-hand and right-hand
derivatives of f exist and increasing on (a; b), and satisfy
¡1 < DL f (x) · DR f (x) < 1
for all x 2 (a; b):
10
Corollary. If f is convex on an open interval (a; b), then f is di®erentiable at all but
countable many points on (a; b); i.e. there is an at most countable set E ½ (a; b) such that
f 0 (x) exists for all x 2 (a; b) n E:
Theorem. Suppsoe f is continuous on a closed interval [a; b] and di®erentiable on (a; b).
If f 0 (x) ¸ 0 for all but countable many x 2 (a; b), then f is increasing on [a; b].
Corollary. Suppsoe f is continuous on a closed interval [a; b] and di®erentiable on (a; b).
If f 0 (x) = 0 for all but countable many x 2 (a; b), then f is constant on [a; b].