B let q n be an enumeration of all the rational

Info icon This preview shows pages 5–6. Sign up to view the full content.

View Full Document Right Arrow Icon
(b) Let <q n > be an enumeration of all the rational numbers in the interval [0,1]. Define f:[0,1] by the following formula: f(x) = 2 -n . q n x Here, of course, the sum is over the indices n such that q n is no larger than x. Note that since the geometric series 2 -n dominates the sum defining f, convergence is not a problem. Show f is continuous at each irrational number in [0,1] and discontinuous at each rational number in [0,1]. Can you find a continuous function g:[0,1] such that the measure of the set {x: f(x) g(x)} is zero? Proof?? Hint: (1) For (a) there are a couple of very easy and obvious functions that do the job. (2) For (b), it helps to observe that the discontinuities are jumps since f is increasing.
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
MAA5616/FINAL EXAM/PART C Autumn, 1997 Page 6 of 6 6. (a) Let f n (x) = n -1 χ [-n,n] (x) for n ε and x ε . Show {f n } converges uniformly to f(x) = 0 on . (b) With proof determine whether there is a Lebesgue integrable function g defined on such the g(x) f n (x) for every x ε and n ε .
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern