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Let $f: [0,1]\to\mathbb{R}$ be a monotonic function.

Show that $f$ is rectificiable. What is the length of $f$?

At first I had to show, that $f$ is at at most countable points not continuous. Now I struggle to show, that $f$ is rectificiable. I doubt that I can go straight by the definition, since I would have to know, what the length $L$ is. But in general I do not know that.

Also I know, that $f$ is riemann-integrable.

Can you give me a hint on how to start here? I am a little bit lost.

Thanks in advance. Hints are appreciated.

Edit:

The definition of a rectificiable curve is as follows:

Let $\gamma: [a,b]\to\mathbb{R}^n$ be a curve. $\gamma$ is rectificiable with length $L$, if for every $\epsilon >0$ it exists a $\delta >0$ such that for every subdivision $a=t_0<\dotso <t_k=b$ with $|t_i-t_{i-1}|<\delta$ is $|\sum_{i=1}^k \|\gamma(t_i)-\gamma(t_{i-1})\|_2-L|<\epsilon$

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Cornman
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  • Also writing down the definition of rectifiable curves would help. – xbh Aug 12 '18 at 13:43
  • @xbh Excuse me, the assumtion, that $f$ is continuous was a typo from me. We only know, that $f$ is monotonic. – Cornman Aug 12 '18 at 13:48
  • Noted. Thanks. I've seen the edit. – xbh Aug 12 '18 at 13:49
  • @xbh I added the definition now. – Cornman Aug 12 '18 at 13:57
  • The sum in your definition hopefully starts at $i=1$ rather than $i=0$. Otherwise the $\gamma(t_{i-1})$ in the first term is meaningless. – hmakholm left over Monica Aug 12 '18 at 14:06
  • @HenningMakholm Yes, you are right. This is not my day, I guess... – Cornman Aug 12 '18 at 14:10
  • If your codomain is $\mathbb R$, then $|\cdot|$ is just the absolute function. By monotonicity, each summand has determined sign and the absolute symbol could be stripped. . – xbh Aug 12 '18 at 14:10
  • @xbh Ok. What I find a little bit odd is, that we defined "curve" as a continuous function. Also I gave now an answer to the question myself. Is that enough to show it is rectificiable and to get the length? Which would be $f(1)-f(0)$ – Cornman Aug 12 '18 at 14:23
  • I would appreciate a comment on my given answer. Is this enough to show that $f$ is rectificiable and to give the length? – Cornman Aug 13 '18 at 08:26

1 Answers1

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Note that you're not being asked for the length of the graph of the function -- but of the length of the curve in $\mathbb R^1$ described by the function.

The first step in the formal part of a solution would indeed be to assert what the length is -- but that doesn't mean that you can't start thinking about what this concept of "length" reduces to in $\mathbb R^1$.

In particular, Hint: Note that when $f$ is monotonic, the $\|{\cdot}\|_2$ signs in your definition are easy to eliminate. And then the sum telescopes!

hmakholm left over Monica
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