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Homework Help: A question on continuous function

  1. Jul 6, 2010 #1
    can anybody please help me in solving the following question:

    consider the function on [0,1] f(x)=1/q if x=p/q, p&q are non zero & p,q are positive integers,
    p/q is in simplest form.
    = 0 if x=0 or irrational

    need to show the set of discontinuities of f(x) is the set of all non zero rationals in [0,1]
    Last edited by a moderator: Jul 7, 2010
  2. jcsd
  3. Jul 6, 2010 #2
    Quick sketch:
    Show that for each real number a, limit of d as x goes to a is 0. You can do that by showing that rationals with bounded numerator are nowhere dense. From this everything you need to prove follows trivially.
    Good luck!
  4. Jul 7, 2010 #3
    Thanks for the help..but I'm still not very clear with your answer. I was thinkng of doing this by using sequential continuity..like takng a random seq. <an> of non zero rationals converging to a non zero rational x. And then claiming <f(an)> not converging to f(x)..so dont know whether this approach is correct..
    Last edited by a moderator: Jul 7, 2010
  5. Jul 7, 2010 #4
    Yes, it's correct. By taking any convergent sequence, we can show that limit of a function is zero at every point. It can be done from Cauchy definition of limit as well. Given a real number x, and arbitrary [tex]\epsilon[/tex], we show that in some neighbour of x rationals have denominators (I've made a mistake in previous post) large enough for function to have values less than [tex]\epsilon[/tex]. Your approach (sequences) is essentially the same.
  6. Jul 7, 2010 #5
    This is not a particularly difficult question, but it most certainly sounds like a question from a class -- either homework or a test question.

    So please explain the origin of the question and why providing you with an approach or an answer is consistent within the ethics of the course -- is not cheating.

    When I took such classes, obtaining outside help on a question like this would have clearly been considered cheating.
    Last edited: Jul 8, 2010
  7. Jul 8, 2010 #6
    this is neithr a homework nor a test question...m doing my msc. now...its a question of riemann integral dat i studied in b.sc...was going thru my notes...v actually need 2 show dis function is riemann integrable bt v cannot use d result"a bdd. functn for which d set of discontnuities has finfitely many limit points is riemann integrable." as dis isnt d case here..my prof. told me d set of continuities of dis fiunction bt i jst wantd 2 verify myself...was getng lil confused...so askd 4 help...jst 2 add 2 my knowledge...dats it..n nt 2 get gud marks or appreciation frm d prof....m nt into all dese thngs...
  8. Jul 8, 2010 #7
    It is Riemann integrable because the set of discontinuities has Lebesgue measure 0.
  9. Jul 8, 2010 #8
    thnx..bt i knew dis function is rieman integrble...was stuck up sumwhere else...
  10. Jul 9, 2010 #9
    thank you for the help..i have got it..
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