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!

Nowhere Continuous Function Dirichlet Proof

  1. Nov 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that the Dirichlet function is continuous nowhere.


    2. Relevant equations
    Dirichlet function = 1 when x is rational, and 0 when x is irrational.


    3. The attempt at a solution
    I was looking at this proof on http://math.feld.cvut.cz/mt/txtd/1/txe4da1c.htm
    At the very end when the creator shows that inf f(x) [tex]\neq[/tex] sup f(x)
    how does this tell you that the function is never continuous?
    Do the greatest lower bound and least upper bound of a function have to be equal at some point for a function to be continuous?
     
  2. jcsd
  3. Nov 30, 2009 #2

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    That's not what he showed.

    He showed the infimum (over all P) of U(f,P) was not the supremum (over all P) of L(f,P).

    And he wasn't proving anything about continuity anyways; that article is about Riemann integrability.
     
  4. Nov 30, 2009 #3
    Oh, wow, don't know how I missed that. Thanks!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Nowhere Continuous Function Dirichlet Proof
Loading...