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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- Nov 1st 2010, 07:53 AMzebra2147Cantor 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. - Nov 1st 2010, 01:20 PMDrexel28
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. - Nov 2nd 2010, 05:49 AMzebra2147
Alright, for continuity, could you say that since there are no isolated points in the Cantor set, for every point $\displaystyle x\in C$ there exists another point $\displaystyle a\in C$ that is arbitrarily close to $\displaystyle x$. Therefore, we can find a $\displaystyle \delta$ such that $\displaystyle |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.