Suppose $a$, $b\in\mathbb{R}$ such that $a<b$, there exists points $a = t_{0} < t_{1} <\cdots<t_{n-1} <t_{n} =b$, and $\gamma\colon[a,b]\to\mathbb{R}^{N}$ is continuous. Then, are the following equivalent, or is there any counterexample?
(1) $\gamma$ is rectifiable.
(2) For all $1\leq i\leq n$, the restriction $\gamma|_{[t_{i-1}, t_{i}]}$ is rectifiable.
If this is true, then $$\sum_{i}\mbox{the length of }\gamma|_{[t_{i-1}, t_{i}]} = \mbox{the length of }\gamma$$ ?
I proved (1)→(2), but couldn’t prove (2)→(1), for partitions of gamma have to be chosen compatibly with each partitions of the restrictions of gamma.
The definition of “rectifiable” and “length”:
Let $\gamma\colon[a, b]\to\mathbb{R}^{N}$ be continuous.
(1) For each partition $P = \{a = x_{0} < x_{1} <\cdots<x_{n-1} <x_{n} =b\}$, define $\Lambda(P, \gamma) = \sum_{i}\|\gamma(x_{i}) - \gamma(x_{i-1})\|$.
(2) $\gamma$ is called rectifiable if $\sup_{P} \Lambda(P, \gamma) < +\infty$.
(3) If $\gamma$ is rectifiable, its length is defined as the supremum.
For (2).
For $a\le a'<b'\le b$ and a partition $Q=\{x_0,...,x_m\}$ of $[a',b']$ with $a'=x_0<...<x_m=b' $ let $G(Q)=\sum_{j=1}^m\|\gamma (x_{j-1})-\gamma (x_j)\|.$ By the Triangle Inequality, if $Q$ and $Q'$ are partitions of $[a',b']$ with $Q\subseteq Q'$ then $G(Q)\le G(Q').$
For $1\le i\le n$ let $L_i$ be the length of $\gamma|_{[t_{i-1},t_i]}\,.$
For any partition $P$ of $[a,b],$ consider the partition $P'=P\cup \{t_0,..,t_n\}.$ For $1\le i\le n,$ observe that $P'_i= P'\cap [t_{i-1},t_i]$ is a partition of $[t_{i-1},t_i],$ so $G(P'_i)\le L_i.$
Also observe that $G(P')=\sum_{i=1}^nG(P'_i).$
Since $P\subseteq P'$ we have $G(P)\le G(P')=\sum_{i=1}^nG(P'_i)\le \sum_{i=1}^nL_i.$