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(Dedekind ��ff�fi
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) fl A,B ffi��� �!�"�#�$�%�&('�)�* 1◦ A∩B = φ;
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7
A 4�1�2�$�3 B /�1�5�$�8
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→ ��� 1 → ��� 2 → ��� 3 → ��� 4 → ��� 5 →
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) fl E ffi�!�"�/�` ( a ) ^�K�#�$�%�8b0 E /�` (a ) ]�^�8
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�
) fl�mEn�o�pfe {In = [an, bn]} '�)�* 1
◦ In ⊃ In+1; 2
◦
lim(bn − an) = 0. 0�q�/�r�m x0 s�t�u /Eo�p In (n ≥ 1).
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��v�w�x�y�z
*f{}| {In}
z�~�
i�j�k�&f�
∞⋂
n=1
In = {x0} .
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e�8
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+}, ε > 0, / n0 ∈
N, � m,n > n0 /
|am − an| < ε.
$�e
{
an = 1−
1
2 + · · ·+ (−1)
n−1 1
n
}
g�h�8
1
+�, p ∈ N .�/
|an+p − an| =
∣∣∣∣ 1n + 1 −
1
n + 2
+ · · ·+ (−1)p−1
1
n + p
∣∣∣∣ .
1
n + 1 −
1
n + 2 + · · ·+(−1)
p−1 1
n + p ������&A0���
=
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u
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1
n + 1
.
n ��2
&
1
n + 1 < ε. «�/ n0 ∈ N, n > n0 �¬
+� p ∈ N /
|an+p − an| < ε.
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{an} ffi�¯�°�e�&f«�g�h�e�8 �
±�²
U�N��W�L�M 5 �CED
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fl A,B ffi}'})
:}<º¹f»
¼Ł¼¼K¼#½$¼%¼8¿¾Àn¼o¼p I0 = [a0, b0] a0
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²
A - B
¹
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¹\Å}Æ�Ç
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Ǻ¹
q}/}m} ½ffi¼¼0¼K½&¿É¼µ I1 = [a1, b1]. Ê}Ë}`}Ì}K}ͼν&¿@¼Ï¼m¼e½¼0Ào
p {In = [an, bn]}. a}Ì}CºD {an}, {bn} ffi Cauchy e}8
+}, ε > 0, / n0 ∈ N,
bn0 − an0 =
b− a
2n0
< ε,
t
ffi� n > n0 & ¬
+� p ∈ N .�/
an0 ≤ an ≤ an+p < bn+p ≤ bn ≤ bn0 .
u
B�/
0 ≤ an+p − an, bn − bn+p < ε.
« {an}, {bn} .�ffi Cauchy e�&
9
L�M 5 @�Ð�& {an}, {bn} /�Ñ�Ò a - b. Ó
9
t
b− a = lim(bn − an) = 0
u
B a = b = α.
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α ∈ A,
©
Q bn ∈ B, u B bn 2 t A
¹
+�Ö
$�&f3 α = lim bn,
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α ffi A
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B
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B /�1�5�$ β, 0 β > α.
t
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2
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{αn},
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l
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1 (
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) fl f(x)
²
[a, b] ¶�·�& f(a)f(b) < 0. 0�q�/ c ∈ (a, b)
f(c) = 0.
���fl f(a) < 0, f(b) > 0.
¾ E = {x| x ∈ [a, b], f(x) > 0}.
©
Q E !�"�/�^�&
®
/�a�]�^}8Gfl inf E = c.
a��CED f(c) = 0.
9
c K ý þ @ Ð &
²
[a, c) ` & f(x) ≤ 0.
u
B
9
¶ · ¸�@�Ð f(c) = f(c−0) ≤ 0.
Ó
©
Q c ffi E K}1}2}a}^}&
®
¬
+¼ n ∈ N, c + 1n ؼffi E K}a}^}& u B}q¼/
c ≤ xn < c +
1
n f(xn) > 0,
9
¶}·}¸}@}Ð f(c + 0) = lim f(xn) ≥ 0.
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q}/
f(c) = 0. �
l
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2 ( ����l
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) fl f(x)
²
[a, b] ¶�·�&f0 f(x) ��¾�� f(a) Ú f(b) �Ep
K
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fl f(x)
²
o�p I `�¶�·�8���C�* J = f(I) Ù�ffi�m� Eo�p\8
fl inf J = a (≥ −∞), supJ = b (≤ +∞).
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a = b, 0 J ffi�m� ����� �K>o�p (�
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% ). a��ff�fl a < b. +�, a < c < b,
9
a, b K�ý�þ�@�Ð�&fq�fi
²
a1, b1 ∈ J
a ≤ a1 < c < b1 ≤ b.
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@�Ð f(x) q�¾�� a1, b1 �>p;K u /��& u B�/ x0 ∈ I f(x0) = c, «
3
c ∈ J .
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²
J
¹
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4 fl f(x)
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Ú�*��/�+�K�,�-�*f�.}& Riemann -�&f`�-�&fa�-�&f`�*�}&¿a�*�
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w�0
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fl f(x)
²
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9
f(x) K�m�%�¶�·�¸}@�Ð}& +�, ε > 0, fi
²
δ > 0,
‖T‖ < δ
�¬
i = 1, 2, · · · , n .�/ 0 ≤ Mi −mi <
ε
b− a
,
t
ffi���
0 ≤ S¯(T )− S (T ) =
n∑
i=1
(Mi −mi)∆xi < ε.
u
B lim
‖T‖→0
(S¯(T )− S (T )) = 0.
9
ª�1
��2�3�!�/
I¯ = I = I =
∫ b
a
f.
l
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1 fl T ′ ffi
9
T 4�û l �
Å
u
µ�K��.�&f0}/
S (T ) ≤ S (T ′) ≤ S (T ) + lω‖T‖,
S¯(T ) ≥ S¯(T ′) ≥ S¯(T )− lω‖T‖.
l
�
2 fl T1 5 T2 ffi [a, b] K��� ��.�&f0�/ S (T1) ≤ S¯(T2).
l
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3 I¯ ≥ I.
§6.3.2
w�0
/��
l
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4
Ô�Õ
I¯ = I = I, 0 f(x)
²
[a, b] @�*�&�6�/
∫ b
a
f = I.
4
l
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5
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f(x)
²
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¬
+�Ö ε > 0, /��. T S¯(T )−S (T ) < ε.
l
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6
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¬
+�, ε > 0, /��. T S¯(T )− S (T ) < ε, 0 I¯ = I.
l
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7 f(x)
²
[a, b] @�*}K}}}q}}Ł¼¼ffi¼* +¼, ε > 0, fi
²
�. T ,
S¯(T )− S (T ) =
n∑
i=1
ωi∆xi < ε.
7�8
1 fl f(x)
²
[a, b] /�^�&fà�9�/}/}Ò;:} ºp�<
Å
&¿0 f(x)
²
[a, b] @
*�8�=�Á�&
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f(x)
²
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7�8
2
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f(x)
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§6.3.3
0
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8
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f(x)
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9 fl f(x) - g(x)
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l
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10
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f(x)
²
[a, b] @�*�&f0
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+� a ≤ c < d ≤ b, f(x)
²
[c, d] @�*�8
l
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11 fl f(x)
²
[a, c] - [c, b] `�@�*�&Ã0 f(x)
²
[a, b] @�*�&?6�/
∫ b
a
f =∫ c
a
f +
∫ a
c
f.
l
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12 fl f(x)
²
[a, b] @�*�& g(x)
²
[a, b] !�¤�c�d�@�&¿0}q}/ ξ ∈ [a, b]
∫ b
a
fg = g(a)
∫ ξ
a
f.
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1
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f(x)
²
[a, b] @�*�& g(x)
²
[a, b] c�d�&f0�q�/ ξ ∈ [a, b]
∫ b
a
fg = g(a)
∫ ξ
a
f + g(b)
∫ b
ξ
f.
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2
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f(x)
²
[a, b] @�*�& g(x)
²
[a, b] !�¤�c�d�ú�& 0�q�/ ξ ∈ [a, b]
∫ b
a
fg = g(b)
∫ b
ξ
f.
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