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# RealVariablesI MATH633

Drexel

GPA 3.63

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This 2 page Class Notes was uploaded by Moshe Swift III on Wednesday September 23, 2015. The Class Notes belongs to MATH633 at Drexel University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/212288/math633-drexel-university in Mathematics (M) at Drexel University.

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Date Created: 09/23/15

Final review problems 1 Construct a bijection between the circle of radius 1 and the segment 07 l 2 Prove that if the distance between any two points of a set E of real numbers is greater than one then the set E is nite or in nite countable 3i Prove that if E is an uncountable set of positive numbers then there exists a number 739 gt 0 such that the set E N 7397 00 is in nite 4i ls the circle C a metric space7 if the distance between two points P E C and Q E C is given by the length of shortest arc of the circle C connecting the points P and Q 5 Prove that the set of all irrational numbers on the real line is a G5 set 6 Let be a function which is increasing and continuous on ab Prove that for any dense subset E of the segment ab the set of points with t E E is dense in fa7 7 Show that the Cantor set is nowhere dense 8i Prove that every open covering of an arbitrary set E on the real line contains a countable subcoveringi 9i ls it possible to construct a closed proper subset of ab which would have measure I 7 a 10 Can the intersection n En of measurable sets where Elegjijan and 00 for every n7 have an in nite measure Can it have a nite measure Zero measure 11 For two measurable subsets A1 and A2 of 07 1 one has mA1 mA2 gt 1 Prove that A1 A2 has a positive measure 12 Prove that an image of a closed unbounded set under a continuous mapping from the real line to itself is not necessarily closed Show that such an image is a set of type Far 13 Let a function de ned on 70000 take only integer values Prove that the set of points of continuity of such a function is open7 and the set of points of discontinuity is close i 14 Are the following functions uniformly continuous a y sinlz on 01 b y zsinlz on 01 15 Are the following statements true a If has bounded variation on ab then has bounded variation on m i b If is continuous and has bounded variation on a b then has bounded variation on a b 16 Prove that if has a derivative at all points of ab then this derivative fz is a measurable function on ab 17 a Show that Df g S Df Dgi b Let f and g be nonnegative and continuous at e Show that Dltf9gt S fCDyC 96Dfc 18 Let fn A f in L177 1 S p lt 007 and let lt9ngt be a sequence of measurable functions such that lgnl S M for all n7 and gn A g are Prove that then gnfn A gf in L17 i 19 Which of the following de nitions of a norm on the linear space of contin uously differentiable functions iiei functions With the continuous derivative on m b is correct a 1118 an WW 3 maxtdmb WOW 0 Mb 1al maxzqa WW 01 WW maxteab WW e ff mm dt mama WW 20 For each a E R nd the norm of the function zt t in all the spaces Lp07 ll S p S 00 Which contain

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