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Nowhere differentiable, continuous

  1. Jun 27, 2007 #1
    Hello
    need help with this one.

    f:[0,1] --> [0,1]

    f( .x1 x2 x3 x4 x5 ...) = .x1 x3 x5 x7

    ( decimal expansion)

    prove that f is nowhere diffrentiable but continuous.
    i tried by just picking a point a in [0,1] and the basic definiton of differentiability about that point...doesnt seem to work! kindly help.
     
  2. jcsd
  3. Jun 27, 2007 #2
    Let x be fixed. You have to show that the limit

    [tex]
    \lim_{y\to x} \frac{f(y)-f(x)}{y-x}
    [/tex]

    does not exist. One way to show this is to choose two different sequences [tex]y_n\to x[/tex] for which the limit

    [tex]
    \lim_{n\to\infty}\frac{f(y_n)-f(x)}{y_n-x}
    [/tex]

    is different. I think I succeeded in choosing two sequences so that one gave 0 as the limit, and other one 1 as the limit, but naturally I'm not throwing all the details here :wink:
     
  4. Jun 27, 2007 #3
    Gladly I didn't show all the details, because my sequence that gave 1 as the limit, had mistakes in it.

    EDIT: I think I got it dealed with. I'll return to the topic if the discussion goes to the details.
     
    Last edited: Jun 27, 2007
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