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סיכום מלא של חדוא 1 א תל אביב
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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)
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[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]
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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)
Heine .A " f . f RA
" 1
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