# Cantor function

• Nov 1st 2010, 07:53 AM
zebra2147
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.
• Nov 1st 2010, 01:20 PM
Drexel28
Quote:

Originally Posted by zebra2147
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.
• Nov 2nd 2010, 05:49 AM
zebra2147
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.