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Math Help - A problem on continuity and differentiability

  1. #1
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    A problem on continuity and differentiability

    Q: Let f:[0,1]->[0,1] be a function defined as follows.
    Take x in [0,1] with the decimal expansion x=.x1x2x3x4x5.....
    Map it to y in [0,1] such that y=.x1x3x5......
    Prove that this map is continuous but nowhere differentiable in [0,1].

    I need some help for this problem.

    If anybody has a solution please mail me at ppcparichoy@gmail.com
    Last edited by ppcparichoy; June 26th 2007 at 04:38 AM.
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  2. #2
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    This is not even a map.

    Consider x=.1111111.... = .1099999....
    Ones gets "mapped" to,
    .1111111...
    The other is "mapped" to,
    .1999999...
    The results are clearly not the same.
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  3. #3
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    No this is not correct

    x=0.111111.... is not equal to 0.1099999.....

    Clearly 1st one is 1/9 and the second one is 11/100.
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  4. #4
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    Quote Originally Posted by ppcparichoy View Post
    x=0.111111.... is not equal to 0.1099999.....

    Clearly 1st one is 1/9 and the second one is 11/100.
    But the same objection applies to x1=0.1100000.. and x2=0.109999.. which
    are equal, but one being mapped to 0.10000.. and the other to 0.19999..=0.2000..,
    which are not equal.

    RonL
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ppcparichoy View Post
    Q: Let f:[0,1]->[0,1] be a function defined as follows.
    Take x in [0,1] with the decimal expansion x=.x1x2x3x4x5.....
    Map it to y in [0,1] such that y=.x1x3x5......
    Prove that this map is continuous but nowhere differentiable in [0,1].

    I need some help for this problem.

    If anybody has a solution please mail me at ppcparichoy@gmail.com
    I think what he is trying to say is that we have a map such that the decimal 0.x1x2x3x4x5..... maps to 0.x1x3x5......

    For example if x = 1 then
    0.1112131415... maps to 0.11315....

    If x = 2 then
    0.2122232425... maps to 0.212325...

    -Dan
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  6. #6
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    Quote Originally Posted by topsquark View Post
    I think what he is trying to say is that we have a map such that the decimal 0.x1x2x3x4x5..... maps to 0.x1x3x5......

    For example if x = 1 then
    0.1112131415... maps to 0.11315....

    If x = 2 then
    0.2122232425... maps to 0.212325...

    -Dan
    Problematic as x \in [0,1],\ x=0.x1x2x3..., the interpretation:

    x \in [0,1],\ x=0.x_1x_2x_3..., with x_i \in \{0,1,2, .. ,9\}

    seems more natural, but it looks ambiguous to me any way.

    RonL
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