《高數(shù)雙語》課件section 1-5

Section1.5ContinuousFunctionsContinuousFunction[連續(xù)函數(shù)]andDiscontinuousPoints[間斷點(diǎn)]2Definition

ConsiderafunctionWesayf(x)iscontinuous[連續(xù)]atx0ifandonlyif

(1)f(x)iswelldefined;(2)Otherwise,wesayf(x)isdiscontinuous[間斷]atx0.Anotherwaytoexpressthedefinitionofcontinuityatx0is:TheContinuityofFunction3iscalledtheincrementofthefunctionalvalue[函數(shù)值增量],orsimplytheincrementofthe

function[函數(shù)增量].Supposethatafunctiony=f(x)isdefinedinWhentheiscalledtheincrementoftheindependentvariable[自變量增量],andYXx0xf(x)f(x0)independentvariablechangesfromx0tox,thecorrespondingfunctionalvaluewillbechangedfromf(x0)tof(x).ThenTheContinuityofFunction4(1)f(x)iswelldefined;(2)f(x)iscontinuousatx0.f(x)iscontinuous[連續(xù)]atx0ifandonlyif

TheContinuityofFunction5TheoremDefinition

(Leftandrightsidecontinuity)

Letx0

D(f),wesayf(x)isleftcontinuous[左連續(xù)]atx0if

andf(x)isrightcontinuous[右連續(xù)]atx0ifSupposethatafunctiony=f(x)isdefinedinthenTheContinuityofFunction6Note

Iff(x)iscontinuousateverypointofanopeninterval

(a,b)thenf(x)issaidtobecontinuousintheinterval(a,b);

iff(x)iscontinuousintheopeninterval(a,b)andiscontinuousfromtherightatpointx=aandisalsocontinuousfromtheleftatx=b,thenf(x)issaidtobecontinuousontheclosedinterval[a,b];iff(x)iscontinuousinanintervalI,thenitiscalledacontinuousfunctioninthisintervalI;iscalledacontinuousfunction.

ifIisjustthedomainofdefinitionofthisfunctionthenf(x)TheContinuityofFunction7Thegraphofacontinuousfunctionisacontinuouscurve.Note

Weusethesymbol

toexpressthesetofall,thatiscontinuousfunctionsintheintervalTheContinuityofFunction8Example

Prove

Proof:

byatrigonometricidentitywehavethenHence

iscontinuousatSince,

isarbitrarypointwehaveFheinterval.ForanyTheContinuityofFunction9Example

Prove

doesnotcontinuousat.Since

andFinish.Proofwehavetheconclusion.TheContinuityofFunction10Example

Determinetheconstantsaandbsuchthatthefollowingfunctioniscontinuousatx=1:Solution:SinceandFromthedefinitionofcontinuity,wecanseethatiff(x)iscontinuousatx=1,thenitmustsatisfywehaveItiseasytoobtaina=2,b=1.Finish.TheClassificationofDiscontinuousPoints11Ifafunctioniscontinuousatx=x0,thenitmustsatisfyallofthefollowingthreeconditions:

and

bothexistandareequal;;

isdefinedat

exists,thatis,.Hence,ifanyoneoftheseconditionsisnotsatisfiedthenthefunctionfisdiscontinuousatx0.Apointatwhichafunctionisdiscontinuousiscalledadiscontinuouspoint[間斷點(diǎn)]ofthefunctionorpointofdiscontinuityofthefunction.TheClassificationofDiscontinuousPoints12FirstkindSecondKindThediscontinuouspointsoffunctioncanbedividedintotwotypes:

at

iscalledadiscontinuityofthefirstIftheleft-sidelimitandright-sidelimitofafunctionthenpointsarecalleddiscontinuityofthesecondkind[第二類間斷點(diǎn)].

bothexist,kind[第一類間斷點(diǎn)]ofthefunction;allotherdiscontinuousTheClassificationofDiscontinuousPoints13Example

Considerthecontinuityofthesignfunctionatx=0.Solution:Sinceandwehaveisadiscontinuityofthejumpdiscontinuouspoint[跳躍間斷點(diǎn)]firstkindofthesignfunction.TheClassificationofDiscontinuousPoints14Solution:Example

Considerthecontinuityofthefunctionat

x=0.Althoughthefunctionisnotdefinedatthepointx=0.isadiscontinuouspointofthefirstkind.ThusxyO1-11removablediscontinuouspoint[可去間斷點(diǎn)]TheClassificationofDiscontinuousPoints15Example

ConsiderthecontinuityofthefunctionatSolution:Sinceisadiscontinuouspointofthesecondkind.infinitediscontinuouspoint[無窮間斷點(diǎn)]TheClassificationofDiscontinuousPoints16Example

Considerthecontinuityofthefunction

atisadiscontinuouspointofthesecondkind.Solution:oscillatingdiscontinuouspoint[振蕩間斷點(diǎn)]Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions17Theorem

Supposethatthefunctionsfandgarebothcontinuousatx=x0.Then (2)

islocallyboundedat.,,()arealsocontinuousat(1);Theorem

Supposethatthefunctioniscomposedfromthefunctions

and,Ifgiscontinuousat

,

and

fiscontinuousatthen

iscontinuousat.Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions18Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions19Example

Provethecontinuityofthetrigonometricfunctions.ExampleProvethecontinuityoftheinversetrigonometricfunctions.Example

Provethecontinuityoftheexponentialfunction.Example

Provethecontinuityofthelogarithmicfunction.Example

Provethecontinuityofthepowerfunction.Alltheelementaryfunctionsarecontinuousintheirdomains.Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions20Example

InvestigatethecontinuityofthefunctionSolution:

except

and.

iscontinuousintheintervalItisclearthatForthepointsince

isajumpdiscontinuouspoint.,Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions21Solution:(continued)

since

isadiscontinuouspointofsecondkind.Forthepoint,Example

InvestigatethecontinuityofthefunctionOperationsoncontinuousfunctionsandthecontinuityofelementaryfunctions22Solution:(continued)

Example

InvestigatethecontinuityofthefunctionOperationsoncontinuousfunctionsandthecontinuityofelementaryfunctions23Note

Bythedefinitionofcontinuityandthecompositeoperationrulewehave

Therefore,whenwecalculatethelimitofacontinuousfunction,theorderoftheoperationsbetweenthelimitandevaluationofthefunctionmaybeinterchanged.Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions24Proof:

canbethoughtofacompositefunctionby

andBythecontinuityof

andthecompositeoperationruleforthelimitExample

ProveSince.wehaveOperationsoncontinuousfunctionsandthecontinuityofelementaryfunctions25Proof:,then,andBythecompositeoperationruleandExample

Prove

as,weobtain

LetOperationsoncontinuousfunctionsandthecontinuityofelementaryfunctions26Proof:Then,and

asHence,SothatExample

Prove,whereLetOperationsoncontinuousfunctionsandthecontinuityofelementaryfunctions27Thelimitofpower-exponentialfunctionsExample

FindSolution:SinceandwehaveFinish.Operationsoncontinuousfunctionsandthecontinuityofelementaryfunctions28Thelimitofpower-exponentialfunctionsExample

FindSolution:SinceandwehaveFinish.Propertiesofcontinuousfunctionsonaclosedinterval29Propertiesofcontinuousfunctionsonaclosedinterval30Proof:wehave,Particularly,wechoose,weobtainthat ,Again,since,wehaveBythelasttheorem,weknownSince,,

suchthat,..,,suchthatPropertiesofcontinuousfunctionsonaclosedinterval31IfweletthenwehaveProof(continued):Finish.Propertiesofcontinuousfunctionsonaclosedinterval32Propertiesofcontinuousfunctionsonaclosedinterval33Propertiesofcontinuousfunctionsonaclosedinterval34Example

Provethatequation

haveatleast