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1、ChapterVIIntegralcalculusinonevariableIntegrationwasinventedforfindingtheareaofshapes.This,ofcourse,isanancientproblem,andthebasicstrategyforsolvingitisequallyold:dividetheshapeintorectanglesandadduptheirareas.Amathematicallysatisfactoryrealizationofthisclear,intuitiv
2、eideaisamaz-inglysubtle.Wenoteinparticularthatisavastnumberofwaysagivenshapecanbeapproximatedbyaunionofrectangles.Itisnotatallself-evidenttheyallleadtothesameresult.Forthisreason,wewillnotdeveloptherigoroustheoryofmeasuresuntilVolumeIII.Inthischapter,wewillconsideron
3、lythesimplercaseofdeterminingtheareabetweenthegraphofasufficientlyregularfunctionofonevariableanditsaxis.Bylayingthegroundworkforapproximatingafunctionbyajuxtaposedseriesofrectangles,wewillseethatthisboilsdowntoapproachingthefunctionbyaseriesofstaircasefunctions,thatis
4、,functionsthatarepiecewiseconstant.Wewillshowthatthisideaforapproximationsisextremelyflexibleandisindependentofitsoriginalgeometricmotivation,andwewillarriveataconceptofintegrationthatappliestoalargeclassofvector-valuedfunctionsofarealvariable.Todeterminepreciselythec
5、lassoffunctionstowhichwecanassignaninte-gral,wemustexaminewhichfunctionscanbeapproximatedbystaircasefunctions.Bystudyingtheconvergenceunderthesupremumnorm,thatis,byaskingifagivenfunctioncanbeapproximateduniformlyontheentireintervalbystaircasefunctions,weareledtothecl
6、assofjumpcontinuousfunctions.Section1isdevotedtostudyingthisclass.There,wewillseethatanintegralisalinearmaponthevectorspaceofstaircasefunctions.Thereisthentheproblemofextendingintegrationtothespaceofjumpcontinuousfunctions;theextensionshouldpreservetheelementaryprope
7、rtiesofthismap,particularlylinearity.Thisexerciseturnsouttobeaspecialcaseofthegeneralproblemofuniquelyextendingcontinuousmaps.Becausetheextensionproblemisofgreatimportanceandentersmanyareasofmathematics,we2VIIntegralcalculusinonevariablewilldiscussitatlengthinSection
8、2.Fromthefundamentalextensiontheoremforuniformlycontinuousmaps,wewillderivethetheoremofcontinuousextensionsofcontinuouslinearmaps.T
