1 "

: http://www.cs.tau.ac.il/~ehudrubi

30/08/2004:

http://www.cs.tau.ac.il/~ehudrubi

a b

[a, b]

a b

(a, b)

a b

[a, b)

a b

(a, b]

)(A

A

x

AxA

A

B

BA

A

B

cAA

BAA BA B

BA BAA B

A B

A - B

(X, Y) = { {x}, {x, y} }

z = (x, y)

x

y

x = Pr1(z)

y = Pr2(z) (x1, y1)(x2, y2)

(x3, y3) (x4, y4)

(x5, y5)y5

y4y3

y1y2

x5x4x3x2x1

G -

Pr1(G)

Pr2(G)

),(),( xyyx

},,{ 321 aaaA =

},,{ 321 bbbB =

=

),(),,(),,(),,(),,(),,(),,(),,(),,(

332313

323212

312111

bababababababababa

BA

ABBA nAAA = ...2AAA =

BBA = )(2PrABA = )(1Pr

(A, B, R) (A, B, R) = (A, B, R)A = AB = BR = R

RyxxRy = ),(

y5y4y3

y1

y2

x5x4x3x2x1

R

AR )(1Pr

BR )(2Pr

(A, A, R)

(A, B, AxB)-

R-1

A BBAf :

R -

zyzx

yx

++

yzxz

zyx

0

x y

z

R

supA AinfA A

AxAAxA

infsup

. , RA

:

. , RA

Dedekind

:

yxByAxRBA

BARBA

z

R , :

.nx>y - n , x>0 - Ryx ,

1][][ + xxx

[x]

.x x V

x)( )(( )

)(

x

)(x

)( x

x

Rx

V(x) x

x)( )(( )

vxxVv )()(),( xVwwvxVv

)()(, 2121 xVvvxVvv

)()()( yVvwyxVwxVv

x)( )(

w

v

y

, " X (V(X X x . X

.

, x x . x

x)( )( ( ) )(

AV- , x V A- x

x)( )(

A

V

y

AcAc

( )

.A0- A A AA 0

. Ac A y .A Ac)0)

.

000

00

)( BABABABA

=

. A , , A=A0 RA

x)( )(

y( )

z( )

A

. .

: x A A x AV

x)( )(

y( )

z( )

A

V

A A A

AA A

AA A = Ac A

. .

. A- x , A x

x )( )( y( ) z( )

A

V

A- A A

='A A

. , x

RAAA '

X A V x . AV

. , A

. x

x -A V x }{xAV =

.

. Z- N

AAA': , A =

Weierstrass-Bolzano

, ). (

RA

)(x

( )

A

: A R . RAIiiEF = )(

Ii

iEA

A

)(( ) ( )( )

E3E2E1

. F F

.F A F . A F

. F- , F- , A- F

, , A F

: , -F (Ei,,Ek) RA

k

iiEA

1=

. Borel-Lebesgue RA

*Nvvn

l

*Nvna >

laal

n

n

= lim .L- an - , an L

. an

. an .

) (

L .

l( )

0|| lan

an , :

00 >*Nvvn > Rl

l

0

*Nvna >

) (

, A an , , A L , . L-

RA

lR

A

an

( )

.A L L an

. A , an Aan lim

},{ += RR

. ) R ( RR

=++=+)(x

x

0>x 0

Weierstrass : - , an

. R

. an

Rl

knala

kn

l

kna

na

( )

an .

Rl

. Cesaro

. , ) ( an +)(

. R. l an , ) ( l an

. -

. -

RnasuplimR

nainflim

: . an

. nn aa supliminflim =

.A- A- .A- A- ,

RA

Cauchy ) (Cauchy an . an

: - 0>

*Nvvn vm

: " ax . x- rn . - a>0 Rx

nrx aa lim=: , x- rn - rn Rx

nnnn rrrr aaa = ''

: - , 0 - nn rr '1' nn rra

nn rr aa limlim ' =

e

xn

ne

nx

=+

)1(lim

bxxe

bxxab

ab

==

==

ln

log

ccara

cacacaac

bb

br

b

bbb

bbb

log)/1(logloglog

loglog)/(logloglog)(log

==

=+=

xb

xbx

xb

b =

=log

)(log

0)ln(

ln >=

=

xxexe

x

x

0ln

0log

>==

>==

yyxey

yyxbyx

bx

mxx

a

am log

loglog =

f . - A x0 -

x0 , W l V* x0 :RRAf :RRl

WxfAVx )(*lxf

xx=

)(lim

0

l

(

(

W

x0( )

V*

A

) ( . x0 - f - A x0 - RAf :R

Cauchy-Bolzano x0 - f . - A x0 -

: x0 *V RAf :R0>

AVyx *,

Stolz

x0( )

A - {x0}

x1 x2 x3 x4 x5x6

xn

)(lim nxfl =

x0 - f . - A x0 - xn - A - {x0} x0 , f(xn) .

RAf :R

Heine l f x0 xn A - {x0} x0 . lxf n )(

x0 - . A x0 -

l f/B - , B x0 - f l . B x0 , :

RAf :AB

x0( )

A

B

l

lxfBxxx

=

)(lim0

) ( . A x0 -

: x0 RAf :lxf

xx=

)(lim

0AB

lxfBxxx

=

)(lim0

!

x0

)(lim0

xfxx

)(lim0

xfxx +

),( 00 xAAx = ),( 00 +=+ xAAx

: , x0 x0 - f

)(lim)(lim)(lim000

xfxfxfxxxxxx +

==

: )(lim),(lim

00

xgxfxxxx

( ) )(lim)( 000

0

0

0

000

00

000

)(lim)(lim

)(lim)(lim

))((lim

)(lim)(lim))((lim

))((lim))((lim

)(lim)(lim))((lim

xg

xx

xg

xx

xx

xx

xx

xxxxxx

xxxx

xxxxxx

xxxfxf

xgxf

xgf

xgxfxfg

xfxf

xgxfxgf

=

=

=

=

+=+

x0 (x0 f .

: x0 (f(x0 W f RRAf :Ax 0

WAVf )(

0x

)( 0xfW

V

AWxfAVf: x0 f A x0 = )}({)( 0

x0 ) ( f .

RAf :Ax 0]),(( 0xA ),[ 0 + xA

) ( f x0 f .

: , x0 RAf :'0 AAx

)()(lim 00

xfxfxx

=

0x

)( 0xf

) ( f x0 .(f(x0 ) ( ,

),[]),(( 00 + xAxA

) ( .

. (f(A , A f , RA

RAf : . f , A

Darboux

c

)(cf

a b

)(af)(bf

. I Darboux I f ) Bolzano( .(f(a)

: Ayx , 0>0>

Heine .A " f . f RA

" 1