Let a curve $C$ in $\mathbb{R}^{2}$ be given by $(x(t), y(t)), t \in[\alpha, \beta] .$ For a partition $\left\{t_{0}, t_{1}, \ldots, t_{n}\right\}$ of $[\alpha, \beta]$, let
$$
\ell(C, P):=\sum_{i=1}^{n} \sqrt{\left[x\left(t_{i}\right)-x\left(t_{i-1}\right)\right]^{2}+\left[y\left(t_{i}\right)-y\left(t_{i-1}\right)\right]^{2}}
$$
If the set $\{\ell(C, P): P$ is a partition of $[\alpha, \beta]\}$ is bounded above, then the curve $C$ is said to be rectifiable, and the length of $C$ is defined to be
$\ell(C):=\sup \{\ell(C, P): P$ is a partition of $[\alpha, \beta]\}$
[Analogous definitions hold for a curve in $\mathbb{R}^{3}$.]
(i) If $\gamma \in(\alpha, \beta)$, and the curves $C_{1}$ and $C_{2}$ are given by $(x(t), y(t))$, $t \in[\alpha, \gamma]$ and by $(x(t), y(t)), t \in[\gamma, \beta]$ respectively, then show that $C$ is rectifiable if and only if $C_{1}$ and $C_{2}$ are rectifiable.
(ii) Suppose that the functions $x$ and $y$ are differentiable on $[\alpha, \beta]$, and one of the derivatives $x^{\prime}$ and $y^{\prime}$ is continuous on $[\alpha, \beta]$, while the other is integrable on $[\alpha, \beta] .$ Show that the curve $C$ is rectifiable and
$$
\ell(C)=\int_{\alpha}^{\beta} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t
$$
(Hint: Propositions $4.18,6.31$, and $3.17$ and Exercise 43 of Chapter
6.) (Compare Exercise 48 of Chapter 6.)
(iii) Show that the conclusion in (ii) above holds if the functions $x$ and $y$ are continuous on $[\alpha, \beta]$ and if there are a finite number of points $\gamma_{0}<\gamma_{1}<\cdots<\gamma_{n}$ in $[\alpha, \beta]$, where $\gamma_{0}=\alpha$ and $\gamma_{n}=\beta$, such that the
assumptions made in (ii) above about the functions $x$ and $y$ hold on each of the subintervals $\left[\gamma_{i-1}, \gamma_{i}\right]$ for $i=1, \ldots, n$
[Note: The result in (iii) above shows that the definition of the length of a piecewise smooth curve given in Section $8.3$ is consistent with the definition of the length of a rectifiable curve given above.]