Final Exam with Solution for Analysis | MATH 554, Exams of Mathematics

Download Final Exam with Solution for Analysis | MATH 554 and more Exams Mathematics in PDF only on Docsity! Prof. Girardi Math 554 Fali 2010 12.10.10 Final Exam ~Heehisegpar |. MARK BOX PROBLEM POINTS 1 2 2 6 3 10 aa sob 3 io NAME: Me . 6 é noth, aR proof - ist choice 10 ~ é 10 proof - rd choice | 10 [ proof th ehoice| 10 | % han | INSTRUCTIONS: (1) This fiual exam is in two parts. This in-class part is closed boak/clesed notes. Once you finish with the in-class part, raise your hand. I will pick wp your in-class part of the exam. and you can stare on the open book/open notes part 2. For this part 2, you need to pick your best 4 proofs out the the 101 gave you earlier and hand these 4 in. (2) The MaRK BOX indicates the problems aloug with their points. This test. is copied 2-sided. Check that your copy of the exam has all of the problems. (3) During this exam, do not leave your seat. If you have a question, raise your hand. When you finish: turn your exam over, put your pencil down, and raise your hand. {4) This exam covers (from Jntroduction to Real Analysis by William F, Trench, Free Edition 1): Sections 1.1, 1.2, 1.3, 4.1, 42, 21,22. Honor Code Statement, {ainderstand that it is the responsibility of every member of the Carolina community to uphold and maintain the University of South Cerolina'y Houor Code. Ax a Carolinian, I certify that I have neither given’ nor reneived unanthorized aid on this exam. Furthermore, | have pot only read but will clso follow the above Instructions. Signature : 1. Let R be the set of real numbers, ® be the extended reel numbers and S cB. La. Atiotber term f ae te. Let S ie a nonempty set that is bounded above. Then the sup 3 is the unique real aumber § & R such that {1} for each 2 & S, (@) #e>G, then 32, & F such that 1d. We imow that sup 5 ¢ ®. More specifically: £. hele @) supS & Rif and only if $ ie mon, Gov (2) sup S = oo if and only if 5 is {3} supS = —so if and only if S is 5 te. Let S be a nonempty set that is bounded below. Phen the inf & such that Ly for each we 8, of SOR , (2) if's > 0, thea Bn, € S such that LE. We know that inf 8 cB. More specifically, below (1) inf 8 ¢ Rif and only if Sis a. a (2) inf S= ~oo # and only if 8 is (2) inh? = 90 if and only if § is  4. Fill in the blanks. In class we showed a sequence converges if and only if it is Cauchy. We did this in 5 steps. (1) Easy direction: A _sequence (5 Catach. 4S‘ eta Gr (2) Lemuna L: A Cauchy sequence is bounded. (3} Corollary 2: A Cauchy sequence has a convergent Congratulations, you have just finished the in-class part. Please raise your hand and [ll pick up your paper and you can start on the take home part. take home part Pick 4 problems from take home part and do them . ' ¢ cf ; 5 art Davendegl atone  & f yak UP ty Met 4 >  function wp: 2 — 3 of a sen T is defined by Inet) a J? net we)=\4! rer. No Show that ay: is continuous at a point 2 € WR if and only if xo € T?U (T°). 8. The characterisi L eynad Prove: Hf f assumes only finitely many values, then_/ is continvons at a poini xo in LE ae Oe Sp Fa * 4 ;. “ DY, ifand only if f is constant on some interval — 8, x5 + 4). «dn be the distinct values of f Suppose fe} = i and a, dy Let Ai, Aa. O<e cminfla Ay li si there is a > O sich that | f(x) ~ f(xo}l < € if |x — xol < 5. Tf f is continuons ai xo, x x — xp] < 8. The converse is obvious. Therefore f(x) = fo) If} ‘ . ; + 4 fx eTand CT (so vat [= (xn 6 Xe 8). Theretore FS T (so xo & T") if x0 jncerval a xy € (TY) if x0 € Te. Conversely. suppose that xo € T°. Then J = (xo — 6.x» +8) C T° tor some 5 > 0. 90 Wa) = 1 for all x © 1, whici implies hat f is continuous at xo. Now suppose eee men ! = bo — & xy +8) C (7)? for some 6 > 0, 90 r(x) = 0 for all , Which again impfies that f i i ce, fi 1 t pode pf J is continuous al xo, Hence, 7 is contirmous at every xq  7. TE {5q)%y is a bounded sequence and limy—oo tin = 0, show that littnc Stn = Us Explain why Trench’s Theorem 4.1.8 (8) cannot be used. 5 Eepus Lauer 8. Find Hmm f(z) and justify your answer with an ¢-6 proot. NS » a=) a) i ; Vhindelee lanai om 4% 1 4 / gs de poms olathe pine Lan 02 Bb po Lani #2 ge eR it  acd ime veo¢ f(r}, £ oxd Js y Me vea+ F(z}, if they exist. Use e-d pzudly, where applicable. your enswers, : : ~ Ea ix- dD ae fx 2 eile ve HAs nor exist. If 2 then (f(x) = Jeet ie ] (ie 2) +3). Therefore Hs mind e2/4) shen FOUL Ve 246. Hence Ji f() = m 10 Recall that lingo f(x) = Lif and only if Tbay—en f(t) = L far cach sequence {eu pe 4 a4 je eS pepad TU