(a) limx→cf(x) =Lif and only if to each>0,there is aδ >0such thatD.(b) limx→cf(x) =Lif and only if for each deleted neighborhoodUofcthere isa neighborhoodVofLsuch thatf(U∩D)⊆V.(c) limx→cf(x) =Lif and only if for every sequence{sn}inDthat converges|f(x)-f(c)|<whenever|x-c|< δ,x∈toc,sn=cfor alln,the sequence{f(sn)}converges toL.(d) limx→cf(x) =Lif and only if limh→0f(c+h) =L.(e) Iffdoes not have a limit atc,then there exists a sequence{sn}inDsn=cfor alln,such thatsn→c, but{f(sn)}diverges.(f) For any polynomialPand any real numberc,limx→cP(x) =P(c).(g) For any polynomialsPandQ,and any real numberc,limx→cP(x)Q(x)=P(c)Q(c).4. Find aδ >0such that 0<|x-3|< δimplies|x2-5x+ 6|<14.5. Find aδ >0such that 0<|x-2|< δimplies|x2+ 2x-8|<110.6. Prove thatlimx→1(4x+ 3) = 7.7. Prove thatlimx→3(x2-2x+ 3) = 6.8. Determine whether or not the following limits exist:(a) limx→0sin1x.(b) limx→0xsin1x.9. Letf:D→Rand letcbe an accumulation point ofD. Suppose that limx→cf(x) =LandL >0. Prove that there is a numberδ >0such thatf(x)>0for allx∈Dwith 0<|x-c|< δ.10.(a) Suppose that limx→cf(x) = 0 and limx→c[f(x)g(x)] = 1. Prove that limx→cg(x)does not exist.27

(b) Suppose thatlimx→cf(x) =L= 0 and limx→c[f(x)g(x)] = 1. Does limx→cg(x)exist, and if so, what is it?III.2CONTINUOUS FUNCTIONSDefinition 3.Letf:D→Rand letc∈D. Thenfis continuous atcif to each>0there is aδ >0such that|f(x)-f(c)|<whenever|x-c|< δ,x∈D.LetS⊆D. Thenfis continuous onSif it is continuous at each pointc∈S.fiscontinuous iffis continuous onD.THEOREM 6. Characterizations of ContinuityLetf:D→Rand letc∈D.The following are equivalent:1.fis continuous atc.2. If{xn}is a sequence inDsuch thatxn→c,thenf(xn)→f(c).3. To each neighborhoodVoff(c),there is a neighborhoodUofcsuch thatf(U∩D)⊆V.Proof:See Theorem 1.CorollaryIfcis an accumulation point ofD,then each of the above is equivalent tolimx→cf(x) =f(c).THEOREM 7.Letf:D→Rand letc∈D. Thenfis discontinuous atcif andonly if there is a sequence{xn}inDsuch thatxn→cbut{f(xn)}does not convergetof(c).Continuity of Combinations of FunctionsTHEOREM 8. Arithmetic:Letf, g:D→Rand letc∈D. Iffandgarecontinuous atc,then1.f+gis continuous atc.2.f-gis continuous atc.3.fgis continuous atc;kfis continuous atcfor any constantk.28

4.f/gis continuous atcprovidedg(c) = 0.THEOREM 9. Composition:Letf:D→Randg:E→Rbe functions such thatf(D)⊆E. Iffis continuous atc∈Dandgis continuous atf(c)∈E,then thecomposition ofgwithf,g◦f:D→R,is continuous atc.Proof:Let>0. Sincegis continuous atf(c)∈Ethere is a positive numberδ1such that|g(f(x))-g(f(c))|<whenever|f(x)-f(c)|< δ1, f(x)∈E. Sincefiscontinuous atcthere is a positive numberδsuch that|f(x)-f(c)|< δ1whenever|x-c|< δ, x∈D. It now follows that|g(f(x))-g(f(c))|<whenever|x-c|< δ, x∈Dandg◦f

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- Fall '08
- Staff
- Topology, Continuity, Limits, lim, Continuous function, Compact space, x→c