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 $|\sum_{i=1}^k \|\gamma(t_i)-\gamma(t_{i-1})\|_2-L|<\epsilon$
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!