PART III. FUNCTIONS: LIMITS AND CONTINUITY

1 PART III. FUNCTIONS: LIMITS AND LIMITS OF FUNCTIONSThis chapter is concerned with functionsf:D RwhereDis a nonempty subset ofR. That is, we will be considering real-valued functions of a real variable. The setDiscalled :D Rand letcbe an accumulation point ofD. A numberLis thelimit offatcif to each >0there exists a >0such that|f(x) L|< wheneverx Dand0<|x c|< .This definition can be stated equivalently as :D Rand letcbe an accumulation point ofD. A numberListhelimit offatcif to each neighborhoodVofLthere exists a deleted neighborhoodUofcsuch thatf(U D) cf(x)= :(a) limx 2(x2 2x+ 4) = 12.(b) limx 2x2 4x 2=4.(c) limx 3x2+3x+5x 3does not exist.(d) limx 1|x 1|x 1does not :Letf(x)=4x 5. Prove that limx 3f(x)= :Let >0.|f(x) 7|=|(4x 5) 7|=|4x 12|=4|x 3|.Choose = /4. Then|f(x) 7|=4|x 3|<4 4= whenever 0<|x 3|< .Two Obvious LIMITS :23(a) For any constantkand any numberc,limx ck=k.

2 (b) For any numberc,limx cx= :D Rand letcbe an accumulation point ofD. Thenlimx cf(x)=Lif and only if for every sequence{sn}inDsuch thatsn c, sn6=cfor alln,f(sn) :Suppose that limx cf(x)=L. Let{sn}be a sequence inDwhich converges toc, sn6=cfor alln. Let >0. There exists >0 such that|f(x) L|< whenever 0<|x c|< (x D).Sincesn cthere exists a positive integerNsuch that|c sn|< for alln> |f(sn) L|< for alln>Nandf(sn) suppose that for every sequence{sn}inDwhich converges toc,f(sn) that limx cf(x)6=L. Then there exists an >0 such that for each >0 there isanx Dwith 0<|x c|< butf(x) L| . In particular, for each positive integernthere is ansn Dsuch that|c sn|<1/nand|f(sn) L| . Now,sn cbut{f(sn)}does not converge toL, a :D Rand letcbe an accumulation point ofD. If limx cf(x) exists,then it is unique. That is,fcan have only one limit :D Rand letcbe an accumulation point cf(x)does not exist, then there exists a sequence{sn}inDsuch thatsn c, but{f(sn)}does not :Suppose that limx cf(x) does not exist.

3 Suppose that for every sequence{sn}inDsuch thatsn c(sn6=c),{f(sn)}converges. Let{sn}and{tn}be sequences inDwhich converge toc. Then{f(sn)}and{f(tn)}are convergent sequences. Let{un}bethe sequence{s1,t1,s2,t2,..}. Then{un}}converges tocand{f(un)}converges tosome numberL. Since{f(sn)}and{f(tn)}are subsequences of{f(un)},f(sn) Landf(tn) L. Therefore, for every sequence{sn}inDsuch thatsn c, sn6=cforalln,f(sn) Land limx cf(x)= of LimitsTHEOREM , g:D Rand letcbe an accumulation point cf(x)=Landlimx cg(x)=M, c[f(x)+g(x)] =L+M, c[f(x) g(x)] =L M, c[f(x)g(x)] =LM,limx c[kf(x)] =kL, kconstant, cf(x)g(x)=LMprovidedM6=0,g(x)6= :(a) Since limx cx=c,limx cxn=cnfor every positive integern, by (3).(b) Ifp(x)=2x3+3x2 5x+ 4, then, by (1), (2) and (3),limx 2p(x)=2( 2)3+3( 2)2 5( 2) + 4 = 10 =p( 2).(c) IfR(x)=x3 2x2+x 5x2+4, then, by (1) (4),limx 2R(x)=23 2(2)2+2 522+4= 38=R(2).THEOREM 4.

4 ( Pinching Theorem )Letf, g, h:D Rand letcbe anaccumulation point ofD. Suppose thatf(x) g(x) h(x)for allx D, x6= cf(x) = limx ch(x)=L,thenlimx cg(x)= :Let >0. There exists a positive number 1such that|f(x) L|< whenever 0<|x c|< 1(x D).That is <f(x) L< whenever 0<|x c|< , there exists a positive number 2such that <h(x) L< whenever 0<|x c|< = min{ 1, 2}. Then <f(x) L g(x) L h(x) L< whenever 0<|x c|< .Therefore, limx cg(x)= Limits25 Definition :D Rand letcbe an accumulation point ofD. A numberLis theright-hand limit offatcif to each >0there exists a >0such that|f(x) L|< wheneverx Dandc<x<c+ .Notation:limx c+f(x)= numberMis theleft-hand limit offatcif to each >0there exists a >0such that|f(x) L|< wheneverx Dandc <x< :limx c f(x)= (a) limx 1 |x 1|x 1= 1; limx 1+|x 1|x 1=1.(b) Letf(x)= x2 1,x 21x 2,x>2;limx 2 f(x)=3,limx 2+f(x) does not cf(x)=Lif and only if each of the one-sided limitslimx c+f(x)andlimx c f(x)exists, andlimx c+f(x) = limx c f(x)= Evaluate the following LIMITS .

5 (a) limx 2x2 4x+3x 1(b) limx 1x2 4x+3x 1(c) limx 2x2 x 6x+2(d) limx 2x2 x 6x+2(e) limx 2x2 x 6(x+2)2(f) limx 1 x 1x 1(g) limx 0x 4+x 2(h) limx 1+1 x2|x 1|2. Given thatf(x)=x3, evaluate the following (a) limx 3f(x) f(3)x 3(b) limx 3f(x) f(2)x 3(c) limx 3f(x) f(2)x 2(d) limx 1f(x) f(1)x 13. True False. Justify your answer by citing a theorem, giving a proof, or giving acounter-example.(a) limx cf(x)=Lif and only if to each >0, there is a >0 such that|f(x) f(c)|< whenever|x c|< , x D.(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 convergestoc,sn6=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}inDsn6=cfor alln, such thatsn c, but{f(sn)}diverges.

6 (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 a >0 such that 0<|x 3|< implies|x2 5x+6|< Find a >0 such that 0<|x 2|< implies|x2+2x 8|< Prove that limx 1(4x+ 3) = Prove that limx 3(x2 2x+ 3) = Determine whether or not the following LIMITS exist:(a) limx 0 sin1x .(b) limx Letf:D Rand letcbe an accumulation point ofD. Suppose that limx cf(x)=LandL>0. Prove that there is a number >0 such thatf(x)>0 for allx Dwith 0<|x c|< .10. (a) Suppose that limx cf(x) = 0 and limx c[f(x)g(x)] = 1. Prove that limx cg(x)does not (b) Suppose that limx cf(x)=L6= 0 and limx c[f(x)g(x)] = 1. Does limx cg(x)exist, and if so, what is it? CONTINUOUS FUNCTIONSD efinition :D Rand letc D. Thenfis continuous atcif to each >0there is a >0such that|f(x) f(c)|< whenever|x c|< , x D. Thenfis continuous onSif it is continuous at each pointc iffis continuous 6.

7 Characterizations of ContinuityLetf:D Rand letc following are continuous If{xn}is a sequence inDsuch thatxn c, thenf(xn) f(c).3. To each neighborhoodVoff(c), there is a neighborhoodUofcsuch thatf(U D) :See Theorem an accumulation point ofD, then each of the above is equivalent tolimx cf(x)=f(c).THEOREM :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 atc, +gis continuous gis continuous continuous atc;kfis continuous atcfor any continuous atcprovidedg(c)6= 9. Composition:Letf:D Randg:E Rbe functions such thatf(D) continuous atc Dandgis continuous atf(c) E, then thecomposition ofgwithf,g f:D R, is continuous :Let >0. Sincegis continuous atf(c) Ethere is a positive number 1such that|g(f(x)) g(f(c))|< whenever|f(x) f(c)|< 1,f(x) E.

8 Sincefiscontinuous atcthere is a positive number such that|f(x) f(c)|< 1whenever|x c|< , x D. It now follows that|g(f(x)) g(f(c))|< whenever|x c|< , x Dandg fis continuous :D R, and letG R. The pre-image ofG, denoted byf 1(G)is the setf 1(G)={x D:f(x) G}.THEOREM functionf:D Ris continuous onDif and only if for each opensetGinRthere is an open setHinRsuch thatH D=f 1(G).Proof:Supposefis continuous onD. LetG Rbe an open set. Ifc f 1(G), thenf(c) G. SinceGis open, there exists a neighborhoodVoff(c) such thatV , there exists a neighborhoodUcofcsuch thatf(Uc D) V. LetH= c f 1(G) open andH D=f 1(G).Conversely, choose anyc D, and letVbe a neighborhood off(c). SinceVis anopen set, there is an open setH Rsuch thatH D=f 1(V). Sincef(c) V, c an open set so there is a neighborhoodUofcsuch thatU (U D) f(H D)= follows thatfis continuous onDby Theorem functionf:R Ris continuous if and only iff 1(G) is open inRwheneverGis open Letf(x)=x2+2x 15x 3.

9 Definefat 3 so thatfwill be continuous at Each of the following functions is defined everywhere except atx= 1. Where possible,definefat 1 so that it becomes continuous at 1.(a)f(x)=x2 1x 1(b)f(x)=1x 1(c)f(x)=x 1|x 1|(d)f(x)=(x 1)2|x 1|3. In each of the following definefat 5 so that it becomes continuous at 5.(a)f(x)= x+4 3x 5(b)f(x)= x+4 3 x 5(c)f(x)= 2x 1 3x 5(d)f(x)= x2 7x+16 6(x 5) x+14. Letf(x)={A2x2,x<2(1 A)x,x what values ofAisfcontinuous at 2?5. Give necessary and sufficient conditions onAandBfor the functionf(x)= Ax B,x 13x,1<x<2Bx2 A,x 2to be continuous atx= 1 but discontinuous atx= Letf:D Rand letc D. True False. Justify your answer by citing adefinition or theorem, giving a proof, or giving a counter-example.(a)fis continuous atcif and only if to each there is a >0 such that|fx) f(c)|< whenever|x c|< andx D.(b) Iff(D) Ris bounded, thenfis continuous onD.(c) Ifcis an isolated point ofD, thenfis continuous atc.}

10 (d) Iffis continuous atcand{xn}is a sequence inD, thenxn cwheneverf(xn) f(c).(e) If{xn}is a Cauchy sequence inD, then{f(xn)}is Prove or give a counterexample.(a) Iffandf+gare continuous onD, thengis continuous onD.(b) Iffandfgare continuous onD, thengis continuous (c) Iffandgare not continuous onD, thenf+gis not continuous onD.(d) Iffandgare not continuous onD, thenfgis not continuous onD.(e) Iff2is continuous onD, thenfis continuous onD.(f) Iffis continuous onD, thenf(D) is a bounded Letf:D R.(a) Prove that iffis continuous atc, then|f|is continuous atc.(b) Suppose that|f|is continuous atc. Does it follow thatfis continuous atc?Justify your Letf:D Rbe continuous atc D. Prove that iff(c)>0, then there is an >0 and a neighborhoodUofcsuch thatf(x)> for allx U Letf:D Rbe continuous atc D. Prove that there exists anM>0 and aneighborhoodUofcsuch that|f(x)| Mfor allx U PROPERTIES OF CONTINUOUS FUNCTIONSD efinition functionf:D Risboundedif there exists a numberMsuch that|f(x)| Mfor allx D.