Analysis 1 Course

Abstract

This manualbringstogethertheanalysis1courseintendedforanofficial program taughttofirst-yearmathematicsandcomputersciencestudents.These are thecoursesthatIpresentedattheFacultyofExactSciencesandComputer Science ofAbdelhamidIbnBadisUniversityofMostaganeminthefirstsemester during theacademicyears2023 − 2025, forfirst-yearstudents.Afterreviewing and reorganizingthecontentofthesecoursesandaddingsomenecessaryconcepts which,inmyopinion,fitwiththecontentoftheprogram.Thismanualwas divided intosevenchapters. The firstchapterlaysthefoundationforunderstandingallthetopicsinthefollo- wing chapters.Initially,wefocusedonbasiclogicalconceptssuchasimplication, converseofanimplication,andthecontrapositiveofanimplication.Wealso presentedsomepatternsproofwithsimpleexamples,aimingtosimplifythemas muchaspossiblesothatstudentscouldbecomefamiliarwithandunderstand these methods. The secondchapterdealswithsomeconceptsthatwewilluselaterincertain mathematical constructions.Inthelatterwehavedefinedtheconcepts:the algebraic propertiesofthecommutativefield(R,+,×) andthatthisfieldis totally ordered,intervals,absolutevaluewiththeseproperties,supremumand infimumofasubsetof R, characterizationofthesupremumandinfimum,the function ofthewholepartofarealnumberandwegivetheArchimedeanproperty finally thedensityofthesetofrationalnumbersin R. The thirdchapterisdevotedtothestudyofsequencesincludingthedefinition of afinitelimitofasequence,convergentsequences,boundedsequences,opera- tions onconvergentsequences,monotonesequencesandsubsequences,recursive sequences andsequencesdefinedbyafunction.Cauchysequencesandtheirpro- pertiesaretreatedindetail.Thechapterconcludeswithdivergentsequencesand their properties. The fourthchapterisalsoimportant,inwhichwedefinearealfunctionofareal variable,thedomainofafunction,theparityofafunction,periodicfunctions, monotonic functions,andseveralotherdefinitionssuchaslocalmaximumand minimumvalues,aswellasboundedfunctions.Weconcludedthischapterby defining afiniteandinfinitelimitatafinitepointandthelimitatinfinity,along with theirproperties. In thefifthchapter,wedealtwiththeconceptofthecontinuityofarealfunc- tion anditsproperties.Severalfundamentaltheoremswerepresentedwiththeir proofs.Weconcludedthischapterwiththeconceptsofcontinuationbyconti- nuity,uniformcontinuity,andLipchitzianfunctions. The lasttwochaptersdealwiththedeferentialityofarealfunctionatapoint and operationsondifferentiablefunctions.Thefundamentaltheoremsondiffe- rentiablefunctionsaregivenindetail.Elementaryfunctionsaretreatedindetail in thelastchapter.