1 D2A3DH 2002 BPCRA0D4ACA3D2A3B9DEA5 BBDG – CFD8BVDE A72003CH6ALA8 A9A1A7BLB0CUAOA4D3A4BADFA6CSAJAW336A815D9A8 A72A8CTf DD[a,∞)CSATC4A2C0DCBKD3A9CJAQx → +∞CVAXBSCK AIC8A2AN integraltext +∞ a f(x)dxD1C5ATALBAADA6DABUD0 integraltext +∞ a xf prime (x)dxA7D1C5A2 D7CYABAMALBAA0 (?=)ATAQCDAUA9CT integraltext +∞ a xf prime (x)dxD1C5A9DIAQ lim x→+∞ xf(x)=0. CZCWCSA9AGAIf AXBSCKAIC8A9BF?f prime (x) ≥ 0,x∈ [a,∞),?AMA9B2CN ACALBAAOATx, A ∈ [a,∞),AFB7CODDA>x>0A9AH | integraldisplay A x xf prime (x)dx| = integraldisplay A x x(?f prime (x))dx ≥ x integraldisplay A x (?f prime (x))dx = x(f(x)?f(A)) ≥ 0. AVACAR lim x→+∞ f(x)=0.CECB | integraldisplay +∞ x xf prime (x)dx| = lim A→+∞ integraldisplay A x x(?f prime (x))dx ≥ lim A→+∞ x(f(x)?f(A)) = xf(x) ≥ 0. BXA9AGA9AG integraltext +∞ a xf prime (x)dxD1C5A9AH 0 = lim x→+∞ | integraldisplay +∞ x xf prime (x)dx|≥ lim x→+∞ xf(x) ≥ 0. BOA9 lim x→+∞ xf(x)=0. B4A1A7BLB0CUAOA4D3A4BADFA6DIAJAW351A87D9A8 7. CTf DD[a, b]×[c,+∞)CSATC4A2B8BCBKD3A9 I(x)= integraldisplay +∞ c f(x, y)dy 2 AM[a, b]CSC4A2A9AQCD I(x)AM[a, b]CSA9ATD1C5A2 DJAQCD B4B9B6B0B1B8(Dini)B2B5AACTC4A2BKD3C7{φ n (x)}B2CCBDx ∈ [a, b] DDAFAOATD3C7A2CQBI lim n→∞ φ n (x)=0,x∈ [a, b]A9CECBA9BKD3C7{φ n (x)} AM[a, b]CSA9ATAVD1C5AI 0A2 AQCDABAGDABU φ 1 (x) ≥ φ 2 (x) ≥ φ n (x) ≥···,x∈ [a, b]. ASA1AQCDABB2AICNBEATε>0A9ANAMAXCLD3N A9CXAS 0 ≤ φ N (x) ≤ ε, x ∈ [a, b]. AHAFB6AQB5ABCQBIANAM ε 0 > 0 A9CXASCNACAXCLD3 n A9B1ANAM x n ∈ [a, b]A9C9AY φ n (x n ) ≥ ε 0 ,n=1,2,···. BRCLA9{x n }AHBWA9ANANAMAWC7{x n k }D1C5A9BQ lim k→∞ x n k = x 0 A2APA5A9 B2CCBDn,ASA6n k >n,BYAH φ n (x n k ) ≥ φ n k (x n k ) ≥ ε 0 AKC3A2D6AAA9CM k → +∞A9AH φ n (x 0 ) = lim k→∞ φ n (x n k ) ≥ ε 0 . APAJDABU lim n→∞ φ n (x 0 )=0CAB3A2 C0C1D5AQCD BBB3B0B1B8 (Dini) B2B5 1. CTB8BCATB4AKBKD3 φ(x, y) AZADAM [a, b]×[c, y 0 )CSA2CQBIφ(x, y)C9AYA71A8B2AICCBDy ∈ [c, y 0 ),AZDDxAT 3 BKD3AM[a, b]CSD0C4A2ATA9A72A8B2CCBDx ∈ [a, b]A9AZDDy ATBKD3AM [c, y 0 )CSAPAYAFAOA9CJ lim y→y ? 0 φ(x, y)=0,x∈ [a, b]A9CECBA9BKD3φ(x, y) AQy → y ? 0 CVBGAI xAM[a, b]CSA9ATAVD1C5AI 0A2 BBB3B0B1B8 (Dini) B2B5 2. CTB8BCATB4AKBKD3 φ(x, y) AZADAM [a, b] × [c,+∞) CSA2CQBI φ(x, y) C9AYA7 1 A8B2AICCBD y ∈ [c,+∞), AZDDxATBKD3AM[a, b]CSD0C4A2ATA9A7 2A8B2CCBDx ∈ [a, b]A9AZDDy ATBKD3AM[c,+∞)CSAPAYAFAOA9CJ lim y→+∞ φ(x, y)=0,x∈ [a, b]A9CECBA9 BKD3φ(x, y)AQy → +∞CVBGAI xAM[a, b]CSA9ATAVD1C5AI 0A2 BBB3B0B1B8(Dini)B2B52B0BCB7AABQφ n (x)=φ(x, n),n=1,2,3···. AGD9CTDABUC0ARA9C4A2BKD3C7{φ n (x)}C9AYBKD3C7ATAUCGAZC2DABUA9 BF{φ n (x)}AM[a, b]CSA9ATAVD1C5AIC8A2?AMA9B2CNBEATε>0,ANAM AXCLD3N, CXASAQn ≥ N CVA9AH 0 ≤ φ n (x) ≤ ε, B2A9CI x ∈ [a, b] AK C3A2 BTAIDABUA71 A8A9AQ y ≥ N CVA9AH 0 ≤ φ(x, y) ≤ φ(x, N) ≤ ε A9B2 A9CIx ∈ [a, b]AKC3AAD6AAA9BKD3φ(x, y)AQy → +∞CVBGAIxAM[a, b] CSA9ATAVD1C5AI 0A2 7 BAB0BCB7ABBQ ψ(x, y)= integraltext y c f(x, t)dt, y ∈ [c,+∞), AABN φ(x, y)= I(x)? integraltext y c f(x, t)dt, y ∈ [c,+∞).CPABBZARA9BKD3ψ(x, y),y∈ [c,+∞),C9 AYDBBHATAUCG(Dini)AZC22A9?AMA9B4AKBKD3φ(x, y)AQy → +∞CVBG AIxAM[a, b]CSA9ATAVD1C5AI0A9BOBJAI?C6BHADBMBA integraltext +∞ c f(x, t)dt AM[a,b]CSA9ATAVD1C5AI I(x)A2