5.8.E: Problems on Absolute Continuity and Rectifiable Arcs
Complete the proofs of Theorems 2 and \(3,\) giving all missing details.
\(\Rightarrow\) Show that \(f\) is absolutely continuous (in the weaker sense) on \([a, b]\) if for every \(\varepsilon>0\) there is \(\delta>0\) such that
\[
\begin{aligned} \sum_{i=1}^{m}\left|f\left(t_{i}\right)-f\left(s_{i}\right)\right|<\varepsilon & \text { whenever } \sum_{i=1}^{m}\left(t_{i}-s_{i}\right)<\delta \text { and } \\ a \leq s_{1} \leq t_{1} \leq s_{2} \leq t_{2} \leq \cdots \leq s_{m} \leq t_{m} \leq b . \end{aligned}
\]
(This is absolute continuity in the stronger sense.)
Prove that \(v_{f}\) is strictly monotone on \([a, b]\) iff \(f\) is not constant on any nondegenerate subinterval of \([a, b] .\)
\(\left.\text { [Hint: If } x<y, V_{f}[x, y]>0, \text { by Corollary 4 of } §7\right] .\)
With \(f, g, h\) as in Theorem 2 of \(§7,\) prove that if \(f, g, h\) are absolutely continuous (in the weaker sense) on \(I,\) so are \(f \pm g, h f,\) and \(|f| ;\) so also is \(f / h\) if \((\exists \varepsilon>0)|h| \geq \varepsilon\) on \(I\).
Prove that
(i) If \(f^{\prime}\) is finite and \(\neq 0\) on \(I=[a, b],\) so also is \(v_{f}^{\prime}(\text { with one-sided }\) derivatives at the endpoints of the interval); moreover,
\[
\left|\frac{f^{\prime}}{v_{f}^{\prime}}\right|=1 \text { on } I .
\]
\(\left.\text { Thus show that } f^{\prime} / v_{f}^{\prime} \text { is the tangent unit vector (see } §1\right)\).
(ii) Under the same assumptions, \(F=f \circ v_{f}^{-1}\) is differentiable on \(J=\left[0, v_{f}(b)\right]\) (with one-sided derivatives at the endpoints of the interval) and \(F[J]=f[I] ;\) i.e., \(F\) and \(f\) describe the same simple arc, with \(V_{F}[I]=V_{f}[I]\).
Note that this is tantamount to a change of parameter. Setting \(s=v_{f}(t),\) i.e., \(t=v_{f}^{-1}(s),\) we have \(f(t)=f\left(v_{f}^{-1}(s)\right)=F(s),\) with the arclength \(s\) as parameter.