matan-1_2

#$"

0

0 1

% 2 # 0

$

/ ( 0' 2 9 *

C [a, b] R Q ; f : [a, b] → R Q

L [a, b] 7 n , , 7 a = x0 < x1 <

· · · < xn−1 < xn = b 7 : , r

[a, b] [xi, xi+1]: i = 0, 1, . . . , n − 1 5

[xi, xi+1] 7 , , ∆xi: max{∆xi : i = 0, . . . , n−

1} = λr ' 7 [xi, xi+1] 7 ,
ξi 7 77	n−1
Sr =	i
	f (ξi)∆xi.
	=0
" f

[a, b] 7 r 7 , 7 , 7 ξi

LC "5"("'$" - - D f

[a, b] ,

	n−1				b
λr →0	i=0	i	i			R
lim		f (ξ )∆x		=	a		,	(1.1)
					f (x)dx = I
, ξi
$, : IR	, : , ε > 0 δ > 0 r {ξi}

(λr < δ |Sr − IR| < ε) ' : , 7 <

7 D % E 5 7

7 D

		LC "5"("'$" -3 C {rk} Q <
rk = {a = x0k							< x1k < · · · < xnkk = b} : , λrk =
max				∆k	} →	0 k	→ ∞	#
1	i	≤	nk{	i
	≤

0 , 2

f [a, b] ,

	nk −1	( i i =	b
				= R
k→∞	r k→∞ i=0
		a	f (x)dx	I ,
lim S k = lim		f ξk)∆xk

rk 7 , D ,

ξik [xki , xki+1]

+C c'"'$" - - 5 M - --3

LC "5"("'$" -2 f : [a, b] → R: <

N- -O:

D 7 D 7 [a, b]

7 , 7 7 R[a, b]

$7 7 " % 7

#"L " - - (f R[a, b]) ( ε > 0 δ > 0 r, r {ξi}, {ξi} (λr, λr < δ |Sr − Sr | < ε)

+C c'"'$" -3 5 7 - -

+ 7 7 < 7

#"L " -3 . f [a, b]

#

C 7: , 7 <

, 7 7 77

n−1

Sr = f (ξi)∆xi,

i=0

7 r " f ,

[a, b]: N 7: [xi, xi+1]O: 7 f

, $7 7

Sr = f (ξi0)∆xi0 + f (ξi)∆xi = f (ξi0)∆xi0 + A.

i=i0

C f , [xi0, xi0+1]: 7 , A: 7 7 7 , , |f (ξi0)|∆xi0: <

D 7 7 , ξi0 [xi0, xi0+1]

|Sr| ≥ |f (ξi0)|∆xi0 − |A|,

0 , 2

f / R[a, b] C , 7 7

+ 7 7 E 7 D

, 7 7 C <

7 7 $

LC "5"("'$" -2 C f : [a, b] → R Q , <

; r = {a = x0 < x1 < · · · < xn = b} Q

[a, b] C 7 mi = inf			f (x): Mi = sup f (x):
[xi,xi+1]			[xi,xi+1]
i = 0, 1, . . . , n − 1
n−1			n−1
Sr = mi∆xi, Sr = Mi∆xi
i=0			i=0

$

L, : Sr ≤ Sr

C r1, r2, r3 Q [a, b] " , <

r1 r2: 7 r1 r2 " 7 , r3 , r1 r2:7 7 r3 = r1 r2

(" - - (r r ) (Sr ≤ Sr ≤

S

r ≤

S

r)

= x0

< x1

C r =

x

< x

<

· · ·

< x

n}

: r

=

x

<

· · ·

<

m

0

{

0

1

m

0

{

0

mn

10

0

x0

0

< · · · < x1

1

< · · ·

−

< xn}

# :

< x1 = x1

< x2 = x2

< xn−1

, :

mj

inf

f x

inf

f

x

m ,

i = x [xij ,xij+1]

(

) ≥ x [xi ,xi+1]

(

) =

i

M 7

n−1

n−1

mj −1

n−1 mj −1

i

∆xij

mij ∆xij = Sr .

Sr = mi∆xi = mi

≥

i=0

=0

j=0

i=0

j=0

# ,<

: ,

(" -3 r1, r2 Sr1 ≤ Sr2

5 : Sr1 ≤ Sr1 r2 ≤ Sr1 r2 ≤ Sr1 .

B ( + 2 3 .

&

0 $' 2

LC "5"("'$" -4

inf Sr = ID sup Sr = ID
r	r

7 $ f <

[a, b]

#"L " -2 f : [a, b] → R # )

' ! ID, ID R

7 7 7 {Sr : r Q } D D D

77 5 f [a, b] ! 77 -3 <

, 77 5 f

		! ,
[	a, b		!I		D =	inf S
	]					r		r

, <

5

/ . # 2 9 *

7 7 5

7 7 7 D 8

A := ( ε > 0 r	(r −		)	(Sr − Sr < ε)).
B := ( ε > 0	δ > 0	r (r −		)
(λr < δ Sr − Sr < ε)).

(" 3- (ID = ID) A

$ ID = ID : , r1, r2 : , ID − ε/2 < Sr1 : ID + ε/2 > Sr2 # r = r1 r2 7 7

ID − ε/2 < Sr1 ≤ Sr ≤ Sr ≤ Sr2 < ID + ε/2,

Sr − Sr < ε

C r Q : A #

Sr ≤ ID ≤ ID ≤ Sr,

ID − ID < ε ' ε > 0 7 7 7:

ID − ID r C M 7 ID = ID (" 3 3 A B

+ A B

0 $' 2

C r Q : A ! 7 δ > 0ε r : ,

	2δ < xi+1 − xi , i = 0, . . . , n − 1, xi r ,
	4	nδM < ε, M	= [a,b]	|	(	)|
			sup		f x	.

# r Q :

r:

λr

< δ # 7 7 Sr − Sr

=

i

(Mi − mi)∆xi +

j

(Mj − mj )∆xj ,

77

7 7

!

, r : 77

7 7

77

2n

, 7

, a:

b: xi r

7 D Q

Q ,

E 7

k

k

7 7 r $7 7

i

(Mi−mi)∆xi

≤

2Mδ·2n < ε.

77

77

=

:

: E D

j

k

, 77 7 D

mj )∆xj

[xk, xk+1]

r $7 7

(Mj

=

k

(Mk

−

mk)∆xk

≤

k

j

−

k

(M

m )

∆xk

(M

Sr < ε.

m )∆x

Sr

<

C M 7 S

S < 2ε

k

− k

r: D λ

k

k

−

k

≤ ≤

k

k ≤ −

r −

r

D

r

δ.

(" 3 2 B (f R[a, b])

$ 3- : , ε > 0 δ > 0 r {ξi} Nr Q

O (λr < δ) 7

7

n−1

i

≤ IR + ε/2.

IR − ε/2 ≤ f (ξi)∆xi

=0

/ D

ξi [xi, xi+1]: , 7

IR − ε/2 ≤ Sr ≤ Sr ≤ IR + ε/2,

,

5 7 5 <

0 ! $ 2

r: B: 7 7

n−1

Sr ≤ f (ξi)∆xi ≤ Sr.

i=0

C B A (ID = ID): Sr ≤ ID = ID ≤ Sr C ID = ID = IR: , 7

n−1

|IR − f (ξi)∆xi| ≤ Sr − Sr < ε.

i=0

$ M D 77

#"L " 3- f : [f, b] → R # )

(f R[a, b]) (ID = ID)

C$" 3- C 7 7 , : <

7 7 5 5 D D(x): <

D D D D , D:

D 5 [0, 1] 1: Q 0 ) < , : 5 D D , [0, 1]: <

7 C M 7 , D 7

C$" 3 3 C f (x) = 1 [a, b] #

	b
a	f (x)dx = a b dx = b − a.
5 :

n−1

Sr = Sr = ∆xi = x1 − a + x2 − x1 + · · · + b − xn−1 = b − a.

i=0

/ 5 * & 9 ' 2 9

7 7 , 7

#"L " 2- (f C[a, b]) (f R[a, b])) E 7 77 5 8

n−1

Sr − Sr = (Mi − mi)∆xi,

i=0

0 ! $ 2

M	sup f (x), m		inf f (x) C f		C	a, b	]
i =	[xi,xi+1]	i =	[xi,xi+1]		[

f C[xi, xi+1]: 7 ! E 7 7 7 , <

xi , xi [xi, xi+1] f (xi) = Mi: f (xi ) =

mi C 7 % 7

ω > 0 7 α > 0 : , x , x [a, b] (|x − x | < α

|f (x ) − f (x )| < ω).

# 7 ε > 0 7 7 δ = min{α, ε/ω}: r Q

: , λr < δ #

		n−1	n−1			ε
S S =		(M m )∆x =	(f (x )	f (x ))∆x < ω			= ε.
	r − r	i		i i	· ω
		i − i i	i −
		=0	i=0

! 77 3 2 , 7 7

#"L " 2 3 * [a, b]

#

C f 7

[a, b] " f (a) = f (b): : M 7

	n−1	n−1
	i
Sr − Sr = (Mi − mi)∆xi = (f (a) − f (a))		∆xi = 0 · (b − a) = 0
=0		i=0

7 r

" f (a) < f (b): ε > 0 7 7

ε

δ < f (b) − f (a),

r Q : λr < δ #

	n−1	n−1
	i
Sr − Sr =		(Mi − mi)∆xi = (f (xi+1) − f (xi))∆xi ≤
=0		i=0

n−1

δ(f (xi+1) − f (xi)) = δ(f (b) − f (a)) < ε.

i=0

0 % 2

/ + 2 9

!L& #!L - N O c [a, b]# ) (f R[a, b])

(f R[a, c] f R[c, b])

	b	c	b
a	f (x)dx = a	f (x)dx + c	f (x)dx.

C f 7 [a, b] # ε > 0 <

7 δ > 0 : , r [a, b] λr < δ

Sr − Sr < ε.

C r Q [a, b]: , r , c # 77 - -

ε> Sr − Sr ≥ Sr − Sr = (Sr − Sr ) + (Sr − Sr ),

Sr (Sr ) Sr (Sr ), Q 77 5 [a, c] ([c, b])

5 ,

- E : 7 :

, a = b 7

a		b	a
	f (x)dx = 0,		f (x)dx = −	f (x)dx.
a		a	b

!L& #!L 3 N O f, g R[a, b] c R# ) f + g R[a, b] cf R[a, b]

		b			b		b
a		(f (x) + g(x))dx = a			f (x)dx + a		g(x)dx,
			b		b
			cf (x)dx = c			f (x)dx.
			a		a
C							7 7 <
7 7
b					n−1
( ( ) +				= λr →0
					i=0	i	i	i
a	f x		g(x))dx lim
					(f (ξ ) + g(ξ ))∆x =

0 % 2
n−1		n−1				b	b
λr →0 i=0	( i i + λr →0 i=0		(	i i
	f ξ )∆x				a		a	g(x)dx.
lim		lim	g ξ )∆x		=	f (x)dx +
b	n−1				n−1			b
		( i	i =
	λr →0 i=0			λr →0	i=0	i i
a		cf ξ )∆x		c lim			= c	a
cf (x)dx = lim					f (ξ )∆x			f (x)dx.

!L& #!L 2 N 7 , O

f, g R[a, b]# ) f g R[a, b] f /g R[a, b] |g(x)| > c > 0[a, b]

L , 7 , Mf,i = sup f (x): mf,i =	inf f (x) 5
[xi,xi+1]	[xi ,xi+1]
D ξ, η [xi, xi+1] 7 7

|f (ξ)g(ξ) − f (η)g(η)| ≤ |f (ξ)| · |g(ξ) − g(η)| + |g(η)| · |f (ξ) − f (η)| ≤
	Kf (Mg,i − mg,i) + Kg(Mf,i − mf,i).
	1		1	=	g(η) − g(ξ)		1	(M m ),
	g(ξ)	− g(η)			g(ξ)g(η)	≤ c2
								g − g

Kf = sup f (x).

x [a,b]

! , D D , , D

ξ, η [xi, xi+1]: 7 D ∆xi 77 i: <

, 7

n−1	n−1			n−1
	i
(Mf g,i−mf g,i)∆xi ≤ Kf	(Mg,i	−mg,i)∆xi+Kg (Mf,i−mf,i)∆xi,
i=0	=0			i=0
n−1		1		n−1
i
(M1/g,i − m1/g,i)∆xi, ≤			c2	(Mg,i − mg,i)∆xi.
=0				i=0

# ε > 0 7 r1 r2 : ,

Sr1 (f ) − Sr1 (f ) < ε, Sr2 (g) − Sr2 (g) < ε.

! , D M 7

r = r1 r2 7 7

Sr(f g) − Sr(f g) < (Kf + Kg)ε,

0 % 2

f, g R[a, b] ! <

,

!L& #!L 4 NTO f, g R[a, b] x [a, b] (f (x) ≤ g(x))

	b	b
	f (x)dx ≤	g(x)dx.
	a	a

NTTO f R[a, b] |f | R[a, b]

b

b

f (x)dx ≤ |f (x)|dx ≤ M(b − a),

a

a

M = sup |f (x)|

x [a,b]

NTO $ 7 7

b		n−1		n−1				b
	λr →0	i=0	( i	i ≤ λr →0 i=0	i	i
a	f (x)dx = lim		f ξ )∆x		g(ξ )∆x		=	a
				lim				g(x)dx.

NTTO C 7 , : , |f | R[a, b] C D ξ, η [a, b] 7 7

||f (ξ)| − |f (η)|| ≤ |f (ξ) − f (η)|,

7 [xi, xi+1] r [a, b] 7 7

E

M i − mi ≤ Mi − mi,

M

=

sup

|

f

x

:

inf

|

f (x) : M m

m

i =

Q <

i

(

)|

x [xi ,xi+1]

|

i

i

x

]

[xi ,xi+1

, f C M 7 D

77 5 |f | D ,

f 8

	n−1								n−1
	i
Sr(|f |) − Sr(|f |) = (Mi −						m	i)∆xi	≤	(Mi − mi)∆xi ≤
=0									i=0
		Sr(f ) − Sr(f ).