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1、ChapterVIIntegralcalculusinonevariableIntegrationwasinventedforfindingtheareaofshapes.This,ofcourse,isanancientproblem,andthebasicstrategyforsolvingitisequallyold:dividetheshapeintorectanglesandadduptheirareas.Amathematicallysatisfactoryrealizationofthisclear,intuitiveide
2、aisamaz-inglysubtle.Wenoteinparticularthatisavastnumberofwaysagivenshapecanbeapproximatedbyaunionofrectangles.Itisnotatallself-evidenttheyallleadtothesameresult.Forthisreason,wewillnotdeveloptherigoroustheoryofmeasuresuntilVolumeIII.Inthischapter,wewillconsideronlythesim
3、plercaseofdeterminingtheareabetweenthegraphofasufficientlyregularfunctionofonevariableanditsaxis.Bylayingthegroundworkforapproximatingafunctionbyajuxtaposedseriesofrectangles,wewillseethatthisboilsdowntoapproachingthefunctionbyaseriesofstaircasefunctions,thatis,functionsth
4、atarepiecewiseconstant.Wewillshowthatthisideaforapproximationsisextremelyflexibleandisindependentofitsoriginalgeometricmotivation,andwewillarriveataconceptofintegrationthatappliestoalargeclassofvector-valuedfunctionsofarealvariable.Todeterminepreciselytheclassoffunctionst
5、owhichwecanassignaninte-gral,wemustexaminewhichfunctionscanbeapproximatedbystaircasefunctions.Bystudyingtheconvergenceunderthesupremumnorm,thatis,byaskingifagivenfunctioncanbeapproximateduniformlyontheentireintervalbystaircasefunctions,weareledtotheclassofjumpcontinuousf
6、unctions.Section1isdevotedtostudyingthisclass.There,wewillseethatanintegralisalinearmaponthevectorspaceofstaircasefunctions.Thereisthentheproblemofextendingintegrationtothespaceofjumpcontinuousfunctions;theextensionshouldpreservetheelementarypropertiesofthismap,particula
7、rlylinearity.Thisexerciseturnsouttobeaspecialcaseofthegeneralproblemofuniquelyextendingcontinuousmaps.Becausetheextensionproblemisofgreatimportanceandentersmanyareasofmathematics,we2VIIntegralcalculusinonevariablewilldiscussitatlengthinSection2.Fromthefundamentalextensio
8、ntheoremforuniformlycontinuousmaps,wewillderivethetheoremofcontinuousextensionsofcontinuouslinearmaps.T
