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I'm trying to construct a fuction described as follows: $f:[0,1]\rightarrow R$ such that $f'(x)=0$ almost everywhere,f has to be continuous and strictly increasing. (I'd also conlude that this functions is not absolutely continuous)
The part in bracket is easy to prove.
I'm in troubles with constructing the function:
I thought about considering the function as an infinite sum of sucession of increasing and conitnuous function, I considered the Cantor-Vitali functions which is defined on $[0,1]$ and is continuous and incresing (not strictly).
So $f(x)=\sum_{k=0}^{\infty} 2^{-k}\phi(3^{-k}x)$ where $2^{-k}$ is there to makes the sum converging and $\phi$ is the Cantor-Vitali function.
The sum convergethe function in continuous (as sum of) and is defined as asked.But I'm in trouble while proving that is strictly increasing.Honestly it seems to be but I don't know how to prove it.
I know that $0\leq x \leq y\leq 1$ always exist a k such that $\phi(3^{-k}x)\leq \phi(3^{-k}y)$ but then I stucked.
I'm looking for help in proving this part.

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See here for other ideas. –  David Mitra Dec 4 '12 at 11:30
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2 Answers

up vote 1 down vote accepted

By $\phi$ we denote Cantor-Vitali function. Let $\{(a_n,b_n):n\in\mathbb{N}\}$ be the set of all intervals in $[0,1]$ with rational endpoints. Define $$ f_n(x)=2^{-n}\phi\left(\frac{x-a_n}{b_n-a_n}\right)\qquad\qquad f(x)=\sum\limits_{n=1}^{\infty}f_n(x) $$ I think you can show that it is continuous a have sero derivative almost everywhere. As for strict monotonicity consider $0\leq x_1<x_2\leq 1$ and find interval $(a_n,b_n)$ such that $(a_n,b_n)\subset(x_1,x_2)$, then $$ f(x_2)-f(x_1)\geq f(b_n)-f(a_n)\geq f_n(b_n)-f(a_n)=2^{-n}>0 $$ So $f$ is strictly monotone.

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Did you mean $n^{-2}$ or $2^{-n}$ I think could work in both ways..or not? –  Laura Dec 4 '12 at 11:38
    
@Laura Both ways, I'll edit my answer to follow your choice. –  Norbert Dec 4 '12 at 11:48
    
Thank you, this approach is more similar to the ones used by my teacher so I think he'd appreciate. –  Laura Dec 4 '12 at 12:17
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In the last formula of Norbert's answer, should it be fn(bn)-fn(an)?

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I realize you do not have enough reputation, but this is really a comment as opposed to an answer. Regards –  Amzoti Apr 9 '13 at 2:35
    
If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. –  Amzoti Apr 9 '13 at 2:36
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