1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sequence of discontinuous functions

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data
    Need an example of a sequence of functions that is discountinuous at every point on [0,1] but converges uniformly to a function that is continuous at every point


    2. Relevant equations



    3. The attempt at a solution
    I used the dirichlet's function as the template
    f_n(x) = 1/n if x is rational and 0 if x is irrational

    f_n(x) is discontinuous at every x in [0,1] and converges to f(x)=0

    But this seems to be a erroneous analysis, because 1/n eventually goes to 0 so f_n(x) will be continuous as n->infinity

    Can i get help in constructing this?
     
  2. jcsd
  3. May 5, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    You already have a good example. What do you mean "because 1/n eventually goes to 0 so f_n(x) will be continuous as n->infinity". Can you give me an example of a value of n where f_n is continuous?
     
  4. May 5, 2009 #3
    Since lim n->inf (1/n)=0, as n-> infinity, f_n(x) will be 0 for rationals as well.

    which means that for any epsilon>0, if n is large enough, |f(x)-0|< epsilon for rational as well?
     
  5. May 5, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Less than epsilon, yes. Equal to zero, no. No f_n is equal to zero. Just because the limit is 0, that doesn't mean f_n becomes zero for any finite n.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Sequence of discontinuous functions
  1. DisContinuous function (Replies: 2)

Loading...