Клевчихин - Матан II семестр

f x

ax

a; b

0; 1

1

1

n

1

1

a

1

a 1

ax dx

lim

k

lim

an

a 1

lim

n

an

.

n

n

1

1

1

1

0

n

1

n

an

n

an

ln a

k

C a; b

a; b

f

ε 0 δ 0 τ : λ τ

δ

0

S f ; τ

s f ; τ

ε

S f ; τ

s f ; τ

n

n

k

xk

τk

ωk

2M

N

N

ωk

xk 2M

xk

2M

ε

N

ε .

4M N

2

k

1

k

1

xk

k

xk

k

ε

2 N 1

N

1

N

1

ωi

xi

ε

ε .

2 N

1

2

k

1 i τk

k

1

ε

f

a; b

f

a; b

f a

f b

f b

f a

f b

f a

a; b

ε

0

0

δ

ε

f b

f a

τ

λ τ

δ

Mk

mk

f xk

f xk 1

n

n

Mk

mk xk

λ τ

f xk

f xk 1

k

1

k

1

δ f x1

f x0

f x2

f x1

f x3

f x2

. . .

f xn 1

f xn 2

f xn

f xn 1

δ f xn f x0

ε

f b

f a

ε.

f b

f a

a

x0 x1

xn b

f

n

0

n

f xk f xk 1

f xk

f xk 1

k

1

k 1

f x1

f x0

f x2

f x1

f x3

f x2

. . .

f xn 1

f xn 2

f xn f xn 1

f xn

f x0

f b f a ,

b

f

f b

f a .

a

f V a; b

F, G

f x F x G x .

F x Vax f

G x F x f x

G

y

x sin 1

0; 1

x

f

a; b

f x dx 0

0

ωk f xk

ωk f xk

ε :

k

1

k

1

αf x

α

0

α 0

ωk αf

sup

αf x

αf x

α

sup

f x

f x

α ωk f .

x ,x

k

x ,x

k

n

ε

ωk

f

xk

α

k

1

n

n

ωk αf xk

α

ωk f xk

α

ε

ε.

α

k

1

k

1

f g

a; b

f

g

f g

a; b

f ; g R a; b

f g

R a; b

f g R a; b .

f g

R a; b

f

g

a; b

f x

M f

g x

M g

ε

0

δ

0

τ

δ

n

n

0

ωk f

k

ε

0

ωk

g

k

ε

.

2M g

2M f

k

1

k 1

f x g x

f x g x

f x g x

f x g x

f x g x

f x g x

f x g x

f x g x

f x g x

f x g x

f x

f x g x

f x g x

g x

M g f x

f x

M f g x

g x .

ωk f g

M g ωk f M f ωk g ,

ε

ε

2M g

2M f

n

n

n

0

ωk f g xk M g

ωk f xk

M f

ωk

g xk

ε

ε

ε.

2

2

k

1

k

1

k 1