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Df HY^ [a, b] C\,F"I, ∀ x ∈ [a, b], Y f ^ x
awf(x)M wf(x) = lim
r→0sup{|f (x1) − f (x2)| : x1, x2∈ (x − r, x + r) ∩ [a, b]}
X+, f ^ x 4T ⇔ wf(x) = 0. ' Dδ = {x ∈ [a, b]|wf(x) ≥ δ}, _ f
4T?LM Df = S∞ n=1
D1 n.
jn (Riemann mkihrql) [a, b]C\,"I f Riemann
1%⇔ Df M6&.
tp (⇒) D f ^ [a, b] CRiemann 1%, δ >0, ∀ ε > 0,
^[a, b]
π: a = x0 < x1<· · · < xn= b,
G
Xk i=1
wi(f ) · ∆xi< ε·δ 2
∗
JVN5, 2007.3
1
`3 wi(x) = sup{|f (x0) − f (x00)| | x0, x00 ∈ [xi−1, xi]}. B x ∈ Dδ ∩ (xi−1, xi), _P@ wi(f ) ≥ wf(x) ≥ δ, Z
X
Dδ∩(xi−1,xi)6=∅
∆xi< ε 2
P@
Dδ⊂ [
Dδ∩(xi−1,xi)6=∅
(xi−1, xi) [n i=0
(xi− ε
4(n + 1), xi+ ε 4(n + 1))
<
X
Dδ∩(xi−1,xi)6=∅
∆xi+ ε
4(n + 1)· 2(n + 1) < ε 2 +ε
2 = ε,
[Y 1, Dδ M6&. `K8, Df = S
n≥1
D1 n
M6&.
(⇐) D |f (x)| ≤ M , ∀ x ∈ [a, b], [)D, Df M6&, ∀ ε > 0,
^0=* {(αi, βi)|i = 1, 2, · · · },GDf ⊂ S
i≥1
(αi, βi),<
X
i≥1
(βi− αi) < ε.
∀ x ∈ [a, b] − S
i≥1
(αi, βi),Zf ^x4T, ^!x 0=*Ix,G
t∈ [a, b] ∩ Ix E
|f (t) − f (x)| < ε.
{(αi, βi), Ix|i ≥ 1, x ∈ [a, b] − S
i≥1
(αi, βi)} M-d&[a, b] W0
, ^\Qf {(αik, βik), Ixl|k = 1, 2, · · · , m, l = 1, 2, · · · , n}. [ LebesgueI2,1> [a, b]
π: a = x0 < x1<· · · < xl= b,
G∀ [xi−1, xi]!]9 (αik, βik) $Ixl e.E Xl
i=1
wi(f )∆xi ≤ X
[xi−1,xi]⊂(αik,βik)
wi(f ) · ∆xi+ X
[xi−1,xi]⊂Ixl
wi(f ) · ∆xi
≤ 2M · ε + 2ε · (b − a)
Zf ^ [a, b]CRiemann 1%. 2