## Question

###### A. $\operatorname{Let} \alpha: I \rightarrow R^{3}$ be a curve of class $C^{0}$ (cf. Exercise 7 ). Use the approximation by polygons described in Exercise 8 to give a reasonable definition of arc length of $\alpha$ b. (A Nonrectifiable Curve.) The following example shows that, with any reasonable definition, the arc length of a $C^{0}$ curve in a closed interval may be unbounded. Let $\alpha:[0,1] \rightarrow R^{2}$ be given as $\alpha(t)=$ $(t, t \sin (\pi / t))$ if $t \neq 0,$ and $\alpha(0)=(

a. $\operatorname{Let} \alpha: I \rightarrow R^{3}$ be a curve of class $C^{0}$ (cf. Exercise 7 ). Use the approximation by polygons described in Exercise 8 to give a reasonable definition of arc length of $\alpha$ b. (A Nonrectifiable Curve.) The following example shows that, with any reasonable definition, the arc length of a $C^{0}$ curve in a closed interval may be unbounded. Let $\alpha:[0,1] \rightarrow R^{2}$ be given as $\alpha(t)=$ $(t, t \sin (\pi / t))$ if $t \neq 0,$ and $\alpha(0)=(0,0) .$ Show, geometrically, that the arc length of the portion of the curve corresponding to $1 /(n+1) \leq$ $t \leq 1 / n$ is at least $2 /\left(n+\frac{1}{2}\right) .$ Use this to show that the length of the curve in the interval $1 / N \leq t \leq 1$ is greater than $2 \sum_{n=1}^{N} 1 /(n+1)$ and thus it tends to infinity as $N \rightarrow \infty$