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Continuity Proof

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that there exist nowhere continuous functions f and g whose sum f+g is continuous on R. Show that the same is true for their product.

    2. Relevant equations

    None

    3. The attempt at a solution

    Let f(x) = 1-D(x), where D(x) is the Dirichlet function
    Let g(x) = D(x)

    (f+g)(x) = 1

    (f*g)(x) = D(x) - D(x)^2 <-- where I'm befuddled

    I know that D(x) can be written as the limit of cos(m!*pi*x)^(2n) as n, m --> infinity and that D(x)^2 is then equal to cos(m!*pi*x)^(4n). Since n --> infinity, are D(x) and D(x)^2 equivalent?
     
  2. jcsd
  3. Mar 13, 2012 #2

    LCKurtz

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    Write out the definitions of f(x) and g(x) in terms of x and look at their product.
     
  4. Mar 13, 2012 #3
    You mean it will alternate between 0^2 and 1^2 then?
     
  5. Mar 13, 2012 #4

    LCKurtz

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    No. I mean if x is rational what are f(x) and g(x) and their product. And what if x is irrational?
     
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