5.7.E: Problems on Total Variation and Graph Length
In the following cases show that \(V_{f}[I]=+\infty,\) though \(f\) is bounded on \(I .(\text { In case (iii), } f \text { is continuous, and in case (iv), it is even differentiable }\) \(\text { on } I .)\)
(i) For \(I=[a, b](a<b), f(x)=\left\{\begin{array}{ll}{1} & {\text { if } x \in R(\text { rational }), \text { and }} \\ {0} & {\text { if } x \in E^{1}-R.}\end{array}\right.\)
(ii) \(f(x)=\sin \frac{1}{x} ; f(0)=0 ; I=[a, b], a \leq 0 \leq b, a<b\).
(iii) \(f(x)=x \cdot \sin \frac{\pi}{2 x} ; f(0)=0 ; I=[0,1]\).
(iv) \(f(x)=x^{2} \sin \frac{1}{x^{2}} ; f(0)=0 ; I=[0,1]\).
[Hints: (i) For any \(m\) there is \(P,\) with
\[
\left|\Delta_{i} f\right|=1, \quad i=1,2, \ldots, m ,
\]
so \(S(f, P)=m \rightarrow+\infty\).
(iii) Let
\[
P_{m}=\left\{0, \frac{1}{m}, \frac{1}{m-1}, \dots, \frac{1}{2}, 1\right\} .
\]
\(\left.\text { Prove that } S\left(f, P_{m}\right) \geq \sum_{k=1}^{m} \frac{1}{k} \rightarrow+\infty .\right]\)
Let \(f : E^{1} \rightarrow E^{1}\) be monotone on each of the intervals
\[
\left[a_{k-1}, a_{k}\right], \quad k=1, \ldots, n \quad\left( \text {"piecewise monotone"}\right) .
\]
Prove that
\[
V_{f}\left[a_{0}, a_{n}\right]=\sum_{k=1}^{n}\left|f\left(a_{k}\right)-f\left(a_{k-1}\right)\right| .
\]
In particular, show that this applies if \(f(x)=\sum_{i=1}^{n} c_{i} x^{i}\) (polynomial), with \(c_{i} \in E^{1}\).
[Hint: It is known that a polynomial of degree \(n\) has at most \(n\) real roots. Thus it is piecewise monotone, for its derivative vanishes at finitely many points (being of \(\text { degree } n-1) . \text { Use Theorem } 1 \text { in } §2 .]\)
\(\Rightarrow\) Prove that if \(f\) is finite and relatively continuous on \(I=[a, b],\) with a bounded derivative, \(\left|f^{\prime}\right| \leq M,\) on \(I-Q(\text { see } §4),\) then
\[
V_{f}[a, b] \leq M(b-a).
\]
However, we may have \(V_{f}[I]<+\infty,\) and yet \(\left|f^{\prime}\right|=+\infty\) at some \(p \in I .\)
\(\text { [Hint: Take } f(x)=\sqrt[3]{x} \text { on }[-1,1] .]\)
Complete the proofs of Corollary 4 and Theorems 2 and 4.
Prove Note 3.
[Hint: If \(|h| \geq \varepsilon\) on \(I,\) show that
\[
\left|\frac{1}{h\left(t_{i}\right)}-\frac{1}{h\left(t_{i-1}\right)}\right| \leq \frac{\left|\Delta_{i} h\right|}{\varepsilon^{2}}
\]
and hence
\[
S\left(\frac{1}{h}, P\right) \leq \frac{S(h, P)}{\varepsilon^{2}} \leq \frac{V_{h}}{\varepsilon^{2}} .
\]
Deduce that \(\frac{1}{h}\) is of bounded variation on \(I\) if \(h\) is. Then apply Theorem 2\((\text { iii) to }\) \(\left.\frac{1}{h} \cdot f .\right]\)
Let \(g : E^{1} \rightarrow E^{1}\) (real) and \(f : E^{1} \rightarrow E\) be relatively continuous on \(J=[c, d]\) and \(I=[a, b],\) respectively, with \(a=g(c)\) and \(b=g(d) .\) Let
\[
h=f \circ g .
\]
Prove that if \(g\) is one to one on \(J,\) then
(i) \(g[J]=I,\) so \(f\) and \(h\) describe one and the same arc \(A=f[I]=h[J]\);
(ii) \(V_{f}[I]=V_{h}[J] ;\) i.e., \(\ell_{f} A=\ell_{h} A\).
[Hint for (ii): Given \(P=\left\{a=t_{0}, \ldots, t_{m}=b\right\},\) show that the points \(s_{i}=g^{-1}\left(t_{i}\right)\) form a partition \(P^{\prime}\) of \(J=[c, d],\) with \(S\left(h, P^{\prime}\right)=S(f, P) .\) Hence deduce \(V_{f}[I] \leq\) \(V_{h}[J] .\)
Then prove that \(V_{h}[J] \leq V_{f}[I],\) taking an arbitrary \(P^{\prime}=\left\{c=s_{0}, \ldots, s_{m}=d\right\}\), \(\left.\text { and defining } P=\left\{t_{0}, \ldots, t_{m}\right\}, \text { with } t_{i}=g\left(s_{i}\right) . \text { What if } g(c)=b, g(d)=a ?\right]\)
Prove that if \(f, h : E^{1} \rightarrow E\) are relatively continuous and one to one on \(I=[a, b]\) and \(J=[c, d],\) respectively, and if
\[
f[I]=h[J]=A
\]
\(\text { (i.e., } f \text { and } h \text { describe the same simple arc } A),\) then
\[
\ell_{f} A=\ell_{h} A .
\]
Thus for simple arcs, \(\ell_{f} A\) is independent of \(f .\)
[Hint: Define \(g : J \rightarrow E^{1}\) by \(g=f^{-1} \circ h .\) Use Problem 6 and Chapter \(4, §9,\) Theorem \(3 .\) First check that Problem 6 works also if \(g(c)=b\) and \(g(d)=a,\) i.e., \(g \downarrow\) \(\text { on } J .]\)
Let \(I=[0,2 \pi]\) and define \(f, g, h : E^{1} \rightarrow E^{2}(C)\) by
\[
\begin{array}{l}{f(x)=(\sin x, \cos x),} \\ {g(x)=(\sin 3 x, \cos 3 x),} \\ {h(x)=\left(\sin \frac{1}{x}, \cos \frac{1}{x}\right) \text { with } h(0)=(0,1).}\end{array}
\]
\(\text { Show that } f[I]=g[I]=h[I] \text { (the unit circle; call it } \mathrm{A}),\) yet \(\ell_{f} A=2 \pi\), \(\ell_{g} A=6 \pi,\) while \(V_{h}[I]=+\infty .\) (Thus the result of Problem 7 fails for \(\text { closed curves and } \text {nonsimple} \text { arcs. })\)
In Theorem \(3,\) define two functions \(G, H : E^{1} \rightarrow E^{1}\) by
\[
G(x)=\frac{1}{2}\left[V_{f}[a, x]+f(x)-f(a)\right]
\]
and
\[
H(x)=G(x)-f(x)+f(a) .
\]
\((G \text { and } H \text { are called, respectively, the positive and negative variation }\) \(\text { functions for } f .)\) Prove that
(i) \(G \uparrow\) and \(H \uparrow\) on \([a, b]\);
(ii) \(f(x)=G(x)-[H(x)-f(a)]\) (thus the functions \(f\) and \(g\) of Theorem 3 are not unique);
(iii) \(V_{f}[a, x]=G(x)+H(x)\);
(iv) if \(f=g-h,\) with \(g \uparrow\) and \(h \uparrow\) on \([a, b],\) then
\[
V_{G}[a, b] \leq V_{g}[a, b], \text { and } V_{H}[a, b] \leq V_{h}\left[a, b\right] ;
\]
(v) \(G(a)=H(a)=0\).
Prove that if \(f : E^{1} \rightarrow E^{n}\left(^{*} C^{n}\right)\) is of bounded variation on \(I=[a, b],\) then \(f\) has at most countably many discontinuities in \(I .\)
[Hint: Apply Problem 5 of Chapter 4, \(§5 .\) Proceed as in the proof of Theorem 4 in
\(\text { §7. Finally, use Theorem 2 of Chapter 1, } §9 .]\)