A continuous function $\alpha: [a,b] \to \mathbb{R}^k$ is called a curve. For each partition $P = \{t_0
Let $l(\alpha) = \sup\{l(\alpha, P): P \text{ partitions } [a,b]\}$
If $l(\alpha) < \infty$, then $\alpha$ is called rectifiable.
a) Let $c < a < b < d.$ Suppose \alpha' is continuous on $(c,d)$ for some curve $\alpha: [c,d] \to \mathbb{R}^k$. Show that the restriction $\alpha|_{[a,b]}$ of $\alpha$ to $[a,b]$ is rectifiable with l(\alpha|_{[a,b]}) = \int_a^b \left|\alpha'(t)\right|\, dt.
b) Let $c < a < b < d.$ Suppose $f: (c,d) \to \mathbb{R}$ is continuously differentiable. Let $\alpha: [a,b] \to \mathbb{R}^2$ be given by $\alpha(t) = (t,f(t))$. Find $l(\alpha)$ in terms of $f$.
Here's what I have so far.
a) we can use the Cauchy-Schwarz inequality, unsure how to implement it, and not sure what else we can invoke b) If $f$ is continuously differentiable, can I talk about uniform convergence, will it help? I am stuck on this one.