матан Бесов - весь 2012

a= ±α0, α1α2 . . .

!

n "

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X Y #

$ x X

$ y Y % "

& X

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x& $ y Y &

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&

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f

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f : X → Y

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# $

% "

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f < g f = g f g f > g f > 0 f 0 f = 0 f = C E

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$

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E

+ ( ", ' ' ( ,

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%

y = c c xα

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! arcsin x arccos x arctg x arcctg x

§

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n

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k=0

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= {−{∞}+∞} ˆ = {∞} U (a) $ % R R R R ε

& ε a ε > 0 U (a) & a ' ! ! Uε(a) ε > 0(!

a

−∞ +∞ ∞

f

˚

Uδ0 (x0) x0 R A R

f x → x0

ε > 0 δ = δ(ε) > 0 : |f (x) − A| < ε

0 < |x − x0| < δ. (2)

+ *

x → a lim f (xn) = A ,

n→∞

{ } ˚ → → ∞ xn xn Uδ0 n N xn a n

% -

' & 1 2

A f x → → a % A f

x → a -

% {xn}

˚ → → ∞

xn Uδ0 (a) xn a n & lim f (xn) = n→∞

= A

$+ ε > 0 # / ,

% % δ = δ(ε) > 0 f (x) Uε(A) x

˚

Uδ (a)

$ * xn → a n → ∞ /

Uε(A) n nδ(ε) f (xn) → A n → ∞

& 2 1 A = lim f (x) x→a

- & A = lim f (x)

x→a

% '

0 {xn}

xn = a, xn → a (n → ∞), f (xn) → A (n → ∞),

A f x → a

- * ,

x → a A R g(x) → A x → a

lim g(x) = A

x→a

! "

{xn}#

f (xn) g(xn) h(xn).

% f (xn) → A h(xn) → A &n → ∞' "

! ( (

g(xn) → A (n → ∞).

) {xn} "

+ ( "

( ( + , -

( ◦

% {xn}

% ( (

lim f (xn)g(xn) = AB ) "

n→∞

{xn}

lim f (x)g(x) = AB

x→a

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f

− f (x )| |f (x ) − A| + |f (x ) − A| < ε + ε = 2ε x , x

˚

Uδ (x0)

% 0- % "

lim f (x) !

x→x0

1 2 &

˚ →

+ (' % xn Uδ0 (x0) xn → x0 n → ∞ ) , ε > 0 %

δ = δ(ε) > 0 0- )

( , nδ(ε) N

{f (x1) f (x2) f (x3) f (x4) . . .}

! "! "! A B # !

$% ! x0

& '

§

x0 R δ > 0 ( Uδ (x0 − 0) = (x0 −

− δ, x0] " x0

δ ) U (x0 − 0) *

x0 &

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x0 δ ) U (x0 + 0)

* x0

&

"

.$

!

.$

/ + & &

$%

x0 − 0 x0 + 0 & x0 R ' & 3

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x0 R f

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x→x0

f (x0 − 0) f (x0 + 0) f (x0 − 0) = f (x0 + 0)

4

§

5 % f X → R "

E X x1, x2 E x1 < x2

f (x1) f (x2) f (x1) f (x2)

6 & &

& $% "

7 *" $% "

. & & *" $% "

b = +∞ +∞ − 0

+∞

sup f = B +∞

(a,b)

ε > 0

! xε (a, b)" f (xε) Uε(B) #$ δ = = δε > 0 % xε Uδ (b) & Uδ (b) ' xε(

˚ −

) f (Uδ (b 0)) Uε(B) ! f * % f (b − 0) = B

+, +

$# +, ! % ' f (a + 0)

f (a, b) x0

f (x0 − 0) f (x0 + 0)

§

a R a

−∞% +∞% x0 − 0% x0 + 0 &x0 R( -! f "

U (a) → R # &

( x → a% lim f (x) = 0 &lim f (x) = ∞(

% $

! + $

!

$ % ! f % g #

˚

U (a)% a R $ a

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lim g(x) = K R K = 0 f g x→a f (x)

x → a

-! f g # +

& ( x → a & #

◦% 3◦

lim g(x) = 1 f g x → a x→a f (x)

§

0 ! * % f U (x0) x0

x0 ' &1 2 '

34

: * 3 , ; 9 $'

&1 2 3 % '

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< ! ! 6 '

1 x * x0 9 1 '

xn * x0" - 6

x = x0 9 xn = x0

! & $"

x0 f '

3 x0" -*

f (x0) = d > 0" > *! ! ε = d2 > 0" ? ( '

δ > 0 |f (x) −

f g x0

f (x) ± g(x)

! " # $ % ! " fg &

x0 f ±g f g

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# ! & " # f, g(

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' ! " f X ! " ϕ + T , ϕ(T ) X - $ ! " &

" " ! " # f ϕ f ◦ ϕ T

! #

(f ◦ ϕ)(t) = f (ϕ(t)), t T.

§

f x0

ϕ t0 ϕ(t0) = x0

f ◦ ϕ t0

' y0 = f (x0) U (y0)

y0 f x0

U (x0) : f (U (x0)) U (y0)

f U (x0)

ϕ t0 U (t0) ϕ(U (t0)) U (x0)

ϕ

U (t0) U (t0) U (x0)

U (t0) !" # f ◦ϕ $

(f ◦ ϕ)(U (t0)) U (y0), % y0 = (f ◦ ϕ)(t0).

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f ◦ ϕ t0 $ '

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f x0

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$

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/ t0 ϕ(t0) = = x0 % ϕ ) t0

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lim (f ◦ ϕ)(t) = y0.

t→t0

f x0 f (x0) = y0

!

" ! # $

% U (x0 + 0) u(x0 − 0) x0 R

[x0, x0 + δ) (x0 − δ, x0] δ > 0

& f U (x0 + 0)

U (x0 −0) "

x0

f (x0 + 0) = f (x0) ( f (x0 − 0) = f (x0)).

f "

x0 ' ' ' " x0

x0 R " x0

x0 x0 "

x0

( x0 " f "

) "

" f (x0 − 0) f (x0 + 0) * + f (x0 + 0) −

−f (x0 −0) " f x0 ,

+ f (x0 + 0) = f (x0 − 0)

x0 "

§

( " ) " -'

"

sgn . / ' 0 1 " -'

+ 2

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§

&

[a, b] " ' "

+ * + "

3 a b "

4 ' "

5 ' f

E E

f (xn) → B n → ∞

{xn} a xnb n N

!"#$! {xnk }

xnk → x0 k → ∞

! # a xnk b #%

x0 [a, b] # f

x0

f (xnk ) → f (x0) k → ∞.

# {f (xnk )} ' % !" ! B ( #

f (xnk ) → B k → ∞.

) # #

sup f = B = f (x0).

[a,b]

*$ # % sup f < +∞

[a,b]

! f # % ! f

x0

+ ! f

#

,

! %

# ! [a, b] %

(a, b)- * ! # %

-

f

[a, b] f (x) > 0 x [a, b] d > 0 f (x) dx [a, b]

# ! f E !

x0 E

f (x) f (x0) x E.

§

, x0 ! f

E . f (x0) ! f

E / ! max f

E

+ !$! f

E f E / min f

E

, # %

! ! ( %

# #

f [a, b] f (a) = A f (b) = B C A B

ξ [a, b] : f (ξ) = C.

0 # ! A = = f (a) C f (b) = B [a, b]

[a1, b1] / #$ # !

f (a1) C f (b1) [a1, b1]

[a2, b2] / #$ # !

f (a2) C f (b2) ! # !$%

"#$! # {[an, bn]} !

f (an) C f (bn).

! # #

f (ξ) C f (ξ) f (ξ) = C,

/

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

ξ (a, b) : f (ξ) = 0.