Riemann DEDCDBDADGDFDD
BJB4BN
BMBA?CDC1CDC7
http://math.nju.edu.cn/?meijq
CYD7 1 BWA? R,BUAW?ε > 0,BBADCMAQAR AAHCRAMBDC1ATBCBP
B6,BZAGCOCBBCBPB6A8AKCVAZCACL ε,CNA9AC5BIA7B2.
D3D9 (1)BDC1B2C0BIA7B2; (2)BIA7AKAHCUB2BTC5BIA7AK; (3)BDC1
ATBIA7AKCQA3BTC5BIA7B2.
BWf C0AJCHCM [a,b]BVCKB8AHBYAYC1,? x ∈ [a,b],AJCHf CMxABAH
CPAPwf(x)C5
wf(x) = limr→0sup{|f(x1)?f(x2)|,x1,x2 ∈ (x?r,x + r)∩[a,b]}
CGB7,f CM x ABBGCC? wf(x) = 0,B3 Dδ = {x ∈ [a,b]|wf(x) ≥ δ},CN f
AHA5BGCCAIBRC4C5 Df =
∞uniontext
n=1
D1
n
.
CYD2 (RiemannD1CZCXCWD6D5D0) [a,b]BVAHCKB8AYC1 f Riemann
BDB1? Df C5BIA7B2.
D8D4 (?) BWf CM[a,b] BVRiemann BDB1,AVAJδ > 0,AL? ε > 0,
ADCM[a,b]AHANAS
pi,a = x0 < x1 < ··· < xn = b,
BZAG
ksummationdisplay
i=1
wi(f)·?xi < ε· δ2
A0C2CEAOC6A1A4AAA6BH,2007.3
1
COBF wi(x) = sup{|f(xprime)? f(xprimeprime)||xprime,xprimeprime ∈ [xi?1,xi]},BUAW x ∈ Dδ ∩
(xi?1,xi),CNC8BSwi(f) ≥ wf(x) ≥ δ,CIAC
summationdisplay
Dδ∩(xi?1,xi)negationslash=?
xi < ε2
C8BS
Dδ?
uniondisplay
Dδ∩(xi?1,xi)negationslash=?
(xi?1,xi)
nuniondisplay
i=0
(xi? ε4(n + 1),xi + ε4(n + 1))
BO
summationdisplay
Dδ∩(xi?1,xi)negationslash=?
xi + ε4(n + 1) ·2(n + 1) < ε2 + ε2 = ε,
CJAJCH1,Dδ C5BIA7B2,COC3BK,Df = uniontext
n≥1
D1
n
C5BIA7B2.
(?) BW|f(x)| ≤ M,? x ∈ [a,b],CJB5BW,Df C5BIA7B2,AU? ε > 0,
ADCMBCBPB6{(αi,βi)|i = 1,2,···},BZAGDf? uniontext
i≥1
(αi,βi),BO
summationdisplay
i≥1
(βi?αi) < ε.
AL? x ∈ [a,b]? uniontext
i≥1
(αi,βi),CIf CMxABBGCC,AUADCMAXxAHBCBPB6Ix,BZ
AGAFt ∈ [a,b]∩Ix BX
|f(t)?f(x)| < ε.
{(αi,βi),Ix|i ≥ 1,x ∈ [a,b]? uniontext
i≥1
(αi,βi)}C5B9CSB2 [a,b]AHCFATBCAQ
AR,AUADCMCKC9CUAQAR {(αik,βik),Ixl|k = 1,2,···,m,l = 1,2,···,n},CJ
LebesgueC1AJBE,BDBQ[a,b]AHANAS
pi,a = x0 < x1 < ··· < xl = b,
BZAG? [xi?1,xi]A2AXCLBL (αik,βik)B0Ixl CT.ACBX
lsummationdisplay
i=1
wi(f)?xi ≤
summationdisplay
[xi?1,xi]?(αik,βik)
wi(f)·?xi +
summationdisplay
[xi?1,xi]?Ixl
wi(f)·?xi
≤ 2M ·ε + 2ε·(b?a)
CIACf CM[a,b]BVRiemannBDB1.
2