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Math Help - Cantor function

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    Cantor function

    I'm studying for an exam and this proof was in my notes. I'm not really sure how to prove it though so I could use some help.

    Proof that the Cantor function is increasing and continuous.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by zebra2147 View Post
    I'm studying for an exam and this proof was in my notes. I'm not really sure how to prove it though so I could use some help.

    Proof that the Cantor function is increasing and continuous.
    What do you think? I agree at first glance the fact that the Cantor function is continuous is a little startling, but write it out. I don't think it's that bad. As for increasing consider the cases as to whether the two numbers contain one in their ternary expansions, etc.
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  3. #3
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    Alright, for continuity, could you say that since there are no isolated points in the Cantor set, for every point x\in C there exists another point a\in C that is arbitrarily close to x. Therefore, we can find a \delta such that |x-a|<\delta \Rightarrow |f(x)-f(a)|<\epsilon . Thus, the definition of continuity holds???

    As for increasing...I haven't quite figured that out yet. It seems slightly easier to accept as truth but I still don't know how to prove it.
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