Матан Лекции

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, %#/ 13 2.

d xx' > 2e ,

) 2 +! / */ ( / x, x¢, ( * ) 3 #0 #" 0. .) #+ )# 0!+, ) $ %#/ f /+ / */ + !+ ' (0, 1), " /, ) , !+ + 2( ' ).

2. % /

. #+ )# # 1 + 0 # # & #) 3 7#" x, x¢, + # 2 #+ )# + 0 # # 3 * )

!" x, x¢. .( 3 !' +!3 ) + 0 2 + * ), &( X /+ / */ 0. .2, ) !+ * $ %##

0 + *# + ' !+ * #. ( 6 / # ( 2# .

1. ) % # f [a, b], f [a, b].

. .+ ( ( 0 * + #+ &. .( 2#, ) $ %#/ f !+ 0 [a, b], /+ / */ + !+ '. -#7 #% # * +#/ + ' !+ * #

$e > 0 "d > 0 $x, x¢ Î [a, b] |x - x¢| < d # |f(x) - f(x¢)| ³ e.

* 0 + 0 2 * #0+ & +!3 d. .2# d = 1, &(

$x1,x¢1 Î [a, b] |x1 -x¢1| < 1 # |f(x1) -f(x¢1)| ³ e.

!3 d = 1/2, &(

(. n- 7 & +!3 d = 1/n, &(

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, %#/ 13 2.

. ( 2# % ** & #) . # & )# (+ * ( + * # x1, x2, ..., xn, ... #x¢1,x¢2, ...,x¢n, ..., ( + + / 6# )#* ! * +#/. . 4 % -' 7 ** + / #0 1 #" (+ " & #) !" * ( + * ' * ( 2# *" (/6 */ ( * ( + * xn1, xn2, ..., xnk, ... . 3 0 )#

lim xn = x0 .

k ®¥ k

f(xn1),f(x¢n1), f(xn2),f(x¢n2), ..., f(xnk),f(x¢nk), ...

*" (# */ f(x0). . # # 7# 1 * ( + * $ ( , ) #+ )# + * +

|f(xnk) -f(x¢nk)| ³ e,

* + ( #+ ( / +* " k ³ 1. .#+ )# * + #+ 2 * *( & ( 2 #/ # 0 )#+ ( 0 * + ! 1.

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, %#/ 13 3. .+ / ' 7 **

3.( 0 '

+ ' !+ * #, *+ '* + !+ * # 0 & # # ( 3 ! ! & 3 !

*+ '* + $ %##. , + #)# # + 1 # 2, # $ %## 3/0 ! 3! & #) ! #, ) 3 ( ( 0 + * ( 6 ' , #0+ * ' ( 0+ # + ' ! ' 7 **.

2. ) % # f [a, b], .

. .* $ %#/ f !+ 0 [a, b]. .2, ) & #) , *

$M "x Î [a, b] |f(x)| £ M.

. + ( ( 0 * + #+ &. .( 2#, ) $ %#/ f & #) . -#7 #% # * +#/ & #) * #

"M $x Î [a, b] |f(x)| > M.

* 0 + 0 2 * #0+ & +!3 M. .2# M = 1, &(

$x1 Î [a, b] |f(x1)| > 1.

!3 M = 2, &(

$x2 Î [a, b] |f(x2)| > 2.

(. n- 7 & +!3 M = n, &(

$xn Î [a, b] |f(xn)| > n.

. ( 2# % ** & #) . # & )# * ( + * x1, x2, ..., xn, ..., ( + + / 6 )#* ! * +#/. . 4 % -' 7 ** 1 & #) / * ( + * * ( 2# *" (/6 */ ( * ( + * xn1, xn2, ..., xnk, ... . 3 0 )#

lim xn = x0 .

k ®¥ k

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, %#/ 13 4. / ' 7 **

. * ( / +* " k ³ 1 * + ( #+! + * +

a £ xnk £ b,

2 + * + * + ( #+! # ( / x0, * x0 Î [a, b].

#, ) $ %#/ f, !+ / 0 [a, b], !+ # + ) x0 Î [a, b]. ( + , ( # ' * ( + *

f(xn1), f(xn2), ..., f(xnk), ...

*" (# */ f(x0) # 1 & #) , ) #+ )# + * +

|f(xnk)| > nk,

* + ( #+ ( / +* " k ³ 1. .#+ )# * + #+ 2 * *( & ( 2 #/ # 0 )#+ ( 0 * + ! 2.

4. 0 0 '

6 ( *+ '* + !+ !" 0 $ %#' +! 2 + * ( 6 ' , 0 + + ' '

' 7 **.

3. ) % # f [a, b], .

. 3 0 )# ) 0 Y 3 0 0 [a, b] # 3 2 ## f. . + ' ' 7 ** $ %#/ f & #) , 1 2 * + Y & #) . .*

a = sup Y, b = inf Y.

!* + 2( #/ ! 3 * * # + , ) a Î Y # b Î Y. 3 + 2( #/ ( 0!+ */ *# #) , 1

+ ( ( 3 ( 0 * + #7 ( / a.

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, %#/ 13 5. , + 2 !" 0 "

. *+ '* + + " ' & # 2 * + * 6 * + * ( + * y1, y2, ..., yn, ... ) #0 2 * + Y, *" (/6 /*/ a. 2( yn Î Y * + * + ) xn Î [a, b] /, ) yn = f(xn). # 2(, & #) / * ( + * x1, x2, ..., xn, ... * ( 2# *" (/6 */ ( * ( + * xn1, xn2, ..., xnk,... . 3 0 )#

f(xn1), f(xn2), ..., f(xnk), ...

*" (# */ f(x0). ( /+ / */ ( * ( + * yn1, yn2, ..., ynk, ... * ( + * # y1, y2, ..., yn, ..., *" (/6 '*/ a. .* ( * ( + * *" (/6 '*/ * ( + * # *" (# */ 2 (,

a = f(x0).

! ( 0 #, ) a /+ / */ 3 0 ) # x0 Î [a, b] # 3 2 ## f # 1 a Î Y, ) 0 + 7 ( 0 * +

! 3.

5. #

), 36 ( / +* " 0 +. 8 $ *# */ ## ( '* +# !" )#*, &( #& 0 + * + ## 0 #) !" ( '. $# 0 + +#(, + * #* 0 + + * ( 6#"

".

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, %#/ 13 5. , + 2 !" 0 "

. !

[a1, b1] É [a2, b2] É ... É [an, bn] É ...

,

lim(bn - an ) = 0.

n®¥

" c ,

c = limn®¥ an = limn®¥ bn.

c2 - c1 = (c2 - bn) + (bn - an) + (an - c1)

+* # * & !" + + ' ) * # /+ / */ 3 * ) ! # # n®¥. .1, " (/ + 1 + * + (

# n®¥, ) + * +

c1 = c2,

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, %#/ 13 5. , + 2 !" 0 "

( 0!+ 6 4.

) +#( , ) ( / 3 & n ³ 1

an £ supn ³ 1 an = c1 = c2 = infn ³ 1 bn £ bn,

1 ) c = c1 = c2 # ( 2# +* 0 [an, bn], n ³ 1.

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#*. 1. !+ / $ %#/, # # 6 / + % +!" ) " 0 ) #/ 0 & 0

# (#+# , ( 0 * + 1 & $ 3 0 )# !" # #) * #" *# #'. ( 6 /

# ( 2# 7#.

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, %#/ 14 1. 7# 2 ) 0 ) ##

1. ) % # f [a, b],

.

. .2( +* & ( + (#, ) * & * + ' ' 7 ** !+ / 0 [a, b] $ %#/ f ( * #& *+ & # # # *#. # 3 0, * 6 * + ) # a1 Î [a, b] # b1 Î [a, b] #, )

f(a1) = m = minx Î [a, b] f(x), f(b1) = M = maxx Î [a, b] f(x).

. ( 2#, #, ) a1 < b1, + #+ * ) + ( *# #) ! ** 2( #/.

!3 #0+ )#* m, 2 ) 2( m # M, m < m < M. 3 */ ( 0, ) '( */ ) c Î [a1, b1], ( / ' f(c) = m. 3 0 $ %#

g(x) = f(x) - m, x Î [a1, b1].

%#/ g !+ [a, b], g(a1) = m - m < 0, g(b1) = M - m > 0. ;* , ) + * + f(c) = m + *# g(c) = 0.

0( # 0 [a1,b1] ) ' (a1 + b1)/2. * #

g ( a1 + b1 ) = 0, 2

#* / ) c = (a1 + b1)/2 '( . #+ * ) #0 (+ " +# [a1, (a1 + b1)/2] # [(a1 + b1)/2, b1] 0 [a1, b1] +!3 , % " ' $ %#/ g # # 0 ) #/ 0 !" 0 +. 3 0 )# +!3 +# ) 0 [a2,

b2].

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, %#/ 14 1. 7# 2 ) 0 ) ##

#*. 2. * %#/ ( 0 * + ! 7# 2 ) 0 ) ##+ 0( # 0 [a2, b2] ) ' (a2 + b2)/2. * #

g ( a2 + b2 ) = 0, 2

bn+1].

129