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Homework Help: Simple dirichlet function differentiability

  1. Nov 25, 2009 #1
    1. The problem statement, all variables and given/known data
    f(x) = {x, x rational, 0, x irrational

    g(x) = {x^2, x rational, 0, x irrational

    Show that f(x) is not differentiable at 0.
    Show that g(x) is differentiable at 0

    2. Relevant equations
    f'(x) = lim(h->0) f(x+h) - f(x)/h I suppose

    3. The attempt at a solution
    Just wondering if I'm thinking right. For f(x) the difference quotient becomes h/0 as h-> 0 and with g(x) it becomes lim(h->0) h^2/h = lim(h->0) h = 0?
  2. jcsd
  3. Nov 25, 2009 #2


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    I suppose you mean h/h?

    That's the general idea, but you have to consider the fact that h may be rational or irrational in a careful writeup.
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