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Homework Help: 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


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    Staff Emeritus
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    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!
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