5.2.E: Problems on Derivatives of Extended-Real Functions
Complete the missing details in the proof of Theorems \(1,2,\) and \(4,\) Corollary \(4,\) and Lemma 1.
\(\text { [Hint for converse to Corollary } 4(\mathrm{ii}) : \text { Use Lemma } 1 \text { for an indirect proof. }]\)
Do cases \(p \leq 0\) in Example \((\mathrm{A})\).
Show that Theorems \(1,2,\) and 4 and Corollaries 2 to 4 hold also if \(f\) is discontinuous at \(a\) and \(b\) but \(f\left(a^{+}\right)\) and \(f\left(b^{-}\right)\) exist and are finite. (In Corollary \(2,\) assume also \(f\left(a^{+}\right)=f\left(b^{-}\right) ;\) in Theorems 1 and 4 and Corollary \(2,\) finiteness is unnecessary.)
\(\text { [Hint: Redefine } f(a) \text { and } f(b) .]\)
Under the assumptions of Corollary \(3,\) show that \(f^{\prime}\) cannot stay infinite on any interval \((p, q), a \leq p<q \leq b .\)
\(\text { [Hint: Apply Corollary } 3 \text { to the interval }[p, q] .]\)
Justify footnote \(1 .\)
[Hint: Let
\[
f(x)=x+2 x^{2} \sin \frac{1}{x^{2}} \text { with } f(0)=0 .
\]
At \(0,\) find \(f^{\prime}\) from Definition 1 in §1. Use also Problem 8 of §1. Show that \(f\) is not \(\left.\text { monotone on any } G_{0}(\delta) .\right]\)
Show that \(f^{\prime}\) need not be continuous or bounded on \([a, b]\) (under the standard metric), even if \(f\) is differentiable there.
\(\text { [Hint: Take } f \text { as in Problem } 5 .]\)
With \(f\) as in Corollaries 3 and \(4,\) prove that if \(f^{\prime} \geq 0\left(f^{\prime} \leq 0\right)\) on \((a, b)\) and if \(f^{\prime}\) is not constantly 0 on any subinterval \((p, q) \neq \emptyset,\) then \(f\) is strictly monotone on \([a, b] .\)
Let \(x=f(t), y=g(t),\) where \(t\) varies over an open interval \(I \subseteq E^{1}\), define a curve in \(E^{2}\) parametrically. Prove that if \(f\) and \(g\) have derivatives on \(I\) and \(f^{\prime} \neq 0,\) then the function \(h=f^{-1}\) has a derivative on \(f[I]\), and the slope of the tangent to the curve at \(t_{0}\) equals \(g^{\prime}\left(t_{0}\right) / f^{\prime}\left(t_{0}\right)\).
\(\text { [Hint: The word "curve" implies that } f \text { and } g \text { are continuous on } I \text { (Chapter } 4, §10),\) so Theorems 1 and 3 apply, and \(h=f^{-1}\) is a function. Also, \(y=g(h(x)) .\) Use \(\text { Theorem } 3 \text { of } §1 .]\)
Prove that if \(f\) is continuous and has a derivative on \((a, b)\) and if \(f^{\prime}\) has a finite or infinite (even one-sided) limit at some \(p \in(a, b),\) then this limit equals \(f^{\prime}(p) .\) Deduce that \(f^{\prime}\) is continuous at \(p\) if \(f^{\prime}\left(p^{-}\right)\) and \(f^{\prime}\left(p^{+}\right)\) exist.
[Hint: By Corollary \(3,\) for each \(x \in(a, b),\) there is some \(q_{x}\) between \(p\) and \(x\) such that
\[
f^{\prime}\left(q_{x}\right)=\frac{\Delta f}{\Delta x} \rightarrow f^{\prime}(p) \text { as } x \rightarrow p .
\]
\(\left.\text { Set } y=q_{x}, \text { so } \lim _{y \rightarrow p} f^{\prime}(y)=f^{\prime}(p) .\right]\)
From Theorem 3 and Problem 8 in §1, deduce the differentiation formulas
\[
(\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}} ;(\arccos x)^{\prime}=\frac{-1}{\sqrt{1-x^{2}}} ;(\arctan x)^{\prime}=\frac{1}{1+x^{2}} .
\]
Prove that if \(f\) has a derivative at \(p,\) then \(f(p)\) is finite, provided \(f\) is not constantly infinite on any interval \((p, q)\) or \((q, p), p \neq q\).
[Hint: If \(f(p)=\pm \infty,\) each \(G_{p}\) has points at which \(\frac{\Delta f}{\Delta x}=+\infty,\) as well as those \(x\) \(\left.\text { with } \frac{\Delta f}{\Delta x}=-\infty .\right]\)