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1、©StephenAbbottMathematicsDepartmentMiddleburyCollegeMiddlebury,VT05753USAabbott@middlebury.eduEditorialBoardS.AxlerF.W.GehringK.A.RibetMathematicsDepartmentMathematicsDepartmentMathematicsDepartmentSanFranciscoStateEastHallUniversityofCaliforniaUniversityUniversityofMichiga
2、natBerkeleySanFrancisco,CA94132AnnArbor,MI48109Berkeley,CA94720-3840USAUSAUSAaxler@sfsu.edufgehring@math.lsa.ribet@math.berkeley.eduumich.eduMathematicsSubjectClassification(2000):2601LibraryofCongressCataloging-in-PublicationDataAbbott,Stephen,1964-Understandinganalysis/St
3、ephenAbbott.p.cm.—(Undergraduatetextsinmathematics)Includesbibliographicalreferencesandindex.1.Mathematicalanalysis.I.Title.II.Series.QA300.A182001515—dc2100-08308ISBN978-1-44I9-2866-5e-ISBN978-0-387-21506-8Printedonacid-freepaper.@2010SpringerScienceBusinessMedia,Inc.Allri
4、ghtsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinfor
5、mationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpression
6、ofopinionastowhetherornottheyaresubjecttoproprietaryrights.PrintedintheUnitedStatesofAmerica.9876springeronline.comSpringerScience+BusinessPrefaceMyprimarygoalinwritingUnderstandingAnalysiswastocreateanelemen-taryone-semesterbookthatexposesstudentstotherichrewardsinherentin
7、takingamathematicallyrigorousapproachtothestudyoffunctionsofarealvariable.Theaimofacourseinrealanalysisshouldbetochallengeandim-provemathematicalintuitionratherthantoverifyit.Thereisatendency,however,tocenteranintroductorycoursetoocloselyaroundthefamiliarthe-oremsofthestand
8、ardcalculussequence.Producingarigorousargumentthatpolynomialsarecontinuousisgoodev
