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.
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.