1ForewordThis is a revised version of Section 7.5 of myAdvanced Calculus(Harper & Row,1978). It is a supplement to my textbookIntroduction to Real Analysis, which is refer-enced several times here. You should review Section 3.4 (Improper Integrals) of thatbook before reading this document.2IntroductionIn Section 7.2 (pp. 462–484) we considered functions of the formF.y/DZbaf .x; y/ dx;cDC4yDC4d:We saw that iffis continuous onOEa; bŁSTXOEc; dŁ, thenFis continuous onOEc; dŁ(Exer-cise 7.2.3, p. 481) and that we can reverse the order of integration inZdcF.y/ dyDZdcZbaf .x; y/ dx!dyto evaluate it asZdcF.y/ dyDZbaZdcf .x; y/ dy!dx(Corollary 7.2.3, p. 466).Here is another important property ofF.Theorem 1Iffandfyare continuous onOEa; bŁSTXOEc; dŁ;thenF.y/DZbaf .x; y/ dx;cDC4yDC4d;(1)is continuously differentiable onOEc; dŁandF0.y/can be obtained by differentiating(1)under the integral sign with respect toyIthat is,F0.y/DZbafy.x; y/ dx;cDC4yDC4d:(2)HereF0.a/andfy.x; a/are derivatives from the right andF0.b/andfy.x; b/arederivatives from the left:ProofIfyandyCSOHyare inOEc; dŁandSOHy¤0, thenF.yCSOHy/NULF.y/SOHyDZbaf .x; yCSOHy/NULf .x; y/SOHydx:(3)From the mean value theorem (Theorem 2.3.11, p. 83), ifx2OEa; bŁandy,yCSOHy2OEc; dŁ, there is ay.x/betweenyandyCSOHysuch thatf .x; yCSOHy/NULf .x; y/Dfy.x; y/SOHyDfy.x; y.x//SOHyC.fy.x; y.x/NULfy.x; y//SOHy:2
From this and (3),ˇˇˇˇˇF.yCSOHy/NULF.y/SOHyNULZbafy.x; y/ dxˇˇˇˇˇDC4Zbajfy.x; y.x//NULfy.x; y/jdx:(4)Now supposeSI > 0. Sincefyis uniformly continuous on the compact setOEa; bŁSTXOEc; dŁ(Corollary 5.2.14, p. 314) andy.x/is betweenyandyCSOHy, there is aı > 0suchthat ifjSOHj< ıthenjfy.x; y/NULfy.x; y.x//j< SI;.x; y/2OEa; bŁSTXOEc; dŁ:This and (4) imply thatˇˇˇˇˇF.yCSOHyNULF.y//SOHyNULZbafy.x; y/ dxˇˇˇˇˇ< SI.bNULa/ifyandyCSOHyare inOEc; dŁand0 <jSOHyj< ı. This implies (2). Since the integralin (2) is continuous onOEc; dŁ(Exercise 7.2.3, p. 481, withfreplaced byfy),F0iscontinuous onOEc; dŁ.