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A c1I0= *,G`S0=*g#R] ε,_ A M6&

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Riemann zxwv|{y

7(;

:.ÆUIUO

http://math.nju.edu.cn/˜meijq

js 1 D A⊂ R, B ∀ε > 0, / ^ A c1I0=

*,G`S0=*g#R] ε,_ A M6&.

ou (1)1I&H6&; (2)6f&AM6; (3) 1I

6bAM6&.

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

(2)

`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

i, βi),<

X

i≥1

i− αi) < ε.

∀ x ∈ [a, b] − S

i≥1

i, βi),Zf ^x 4T, ^!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

i, 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 ik, βik) $Ixl e. E Xl

i=1

wi(f )∆xi X

[xi−1,xi]⊂(αikik)

wi(f ) · ∆xi+ X

[xi−1,xi]⊂Ixl

wi(f ) · ∆xi

≤ 2M · ε + 2ε · (b − a)

Z f ^ [a, b]CRiemann 1%. 2

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