6.7.E: Problems on Inverse and Implicit Functions, Open and Closed Maps

Exercise $\PageIndex{1}$

Discuss: In Definition 1, $\overline{G}$ can equivalently be replaced by $G=G_{\vec{p}}(\delta)$ (an open globe).

Exercise $\PageIndex{2}$

Prove that if the set $D$ is open (closed) in $(S, \rho),$ then the map $f : S \rightarrow T$ is open (closed, respectively) on $D$ iff $f_{D}(f \text { restricted to } D)$ has this property as a map of $D$ into $f[D]$.
[Hint: Use Theorem 4 in Chapter 3, §12.]

Exercise $\PageIndex{3}$

Complete the missing details in the proofs of Theorems 1-4.

Exercise $\PageIndex{3'}$

Verify footnotes 2 and 3.

Exercise $\PageIndex{4}$

Show that a map $f : E^{\prime} \rightarrow E$ may fail to be one-to-one on all of $E^{\prime}$ even if $f$ satisfies Theorem 1 near every $\vec{p} \in E^{\prime}.$ Nonetheless, show that this cannot occur if $E^{\prime}=E=E^{1}$.
[Hints: For the first part, take $E^{\prime}=C, f(x+i y)=e^{x}(\cos y+i \sin y).$ For the second, use Theorem 1 in Chapter 5, §2.]

Exercise $\PageIndex{4'}$

(i) For maps $f : E^{1} \rightarrow E^{1},$ prove that the existence of a bijective $d f(p ; \cdot)$ is equivalent to $f^{\prime}(p) \neq 0$.
(ii) Let
$$f(x)=x+x^{2} \sin \frac{1}{x}, \quad f(0)=0.$$
Show that $f^{\prime}(0) \neq 0,$ and $f \in C D^{1}$ near any $p \neq 0;$ yet $f$ is not one-to-one near 0. What is wrong?

Exercise $\PageIndex{5}$

Show that a map $f : E^{n}\left(C^{n}\right) \rightarrow E^{n}\left(C^{n}\right), f \in C D^{1},$ may be bijective even if $\operatorname{det}\left[f^{\prime}(\vec{p})\right]=0$ at some $\vec{p},$ but then $f^{-1}$ cannot be differentiable at $\vec{q}=f(\vec{p}).$
[Hint: For the first clause, take $f(x)=x^{3}, p=0;$ for the second, note that if $f^{-1}$ is differentiable at $\vec{q},$ then Note 2 in §4 implies that det $[d f(\vec{p} ; \cdot)] \cdot \operatorname{det}\left[d f^{-1}(\vec{q} ; \cdot)\right]=1 \neq 0,$ since $f \circ f^{-1}$ is the identity map.]

Exercise $\PageIndex{6}$

Prove Corollary 2 for the general case of complete $E^{\prime}$ and $E$.
[Outline: Given a closed $X \subseteq \overline{G},$ take any convergent sequence $\left\{\vec{y}_{n}\right\} \subseteq f[X].$ By Problem 8 in Chapter 4, §8, $f^{-1}\left(\vec{y}_{n}\right)=\vec{x}_{n}$ is a Cauchy sequence in $X$ (why?). By the completeness of $E^{\prime},(\exists \vec{x} \in X) \vec{x}_{n} \rightarrow \vec{x}$ (Theorem 4 of Chapter 3, §16). Infer that $\lim \vec{y}_{n}=f(\vec{x}) \in f[X],$ so $f[X]$ is closed.]

Exercise $\PageIndex{7}$

Prove that "the composite of two open (closed) maps is open (closed)." State the theorem precisely. Prove it also for the uniform Lipschitz property.

Exercise $\PageIndex{8}$

Prove in detail that $f :(S, \rho) \rightarrow\left(T, \rho^{\prime}\right)$ is open on $D \subseteq S$ iff $f$ maps the interior of $D$ into that of $f[D];$ that is, $f\left[D^{0}\right] \subseteq(f[D])^{0}$.

Exercise $\PageIndex{9}$

Verify by examples that $f$ may be:
(i) closed but not open;
(ii) open but not closed.
[Hints: (i) Consider $f=$ constant. (ii) Define $f : E^{2} \rightarrow E^{1}$ by $f(x, y)=x$ and let
$$D=\left\{(x, y) \in E^{2} | y=\frac{1}{x}, x>0\right\};$$
use Theorem 4(iii) in Chapter 3, §16 and continuity to show that $D$ is closed in $E^{2},$ but $f[D]=(0,+\infty)$ is not closed in $E^{1}.$ However, $f$ is open on all of $E^{2}$ by Problem 8. (Verify!)]

Exercise $\PageIndex{10}$

Continuing Problem 9(ii), define $f : E^{n} \rightarrow E^{1}$ (or $C^{n} \rightarrow C)$ by $f(\vec{x})=$ $x_{k}$ for a fixed $k \leq n$ (the "$k$th projection map"). Show that $f$ is open, but not closed, on $E^{n}\left(C^{n}\right)$.

Exercise $\PageIndex{11}$

(i) In Example (a), take $(p, q)=(5,0)$ or $(-5,0).$ Are the conditions of Theorem 4 satisfied? Do the conclusions hold?
(ii) Verify Example (b).

Exercise $\PageIndex{12}$

(i) Treating $z$ as a function of $x$ and $y,$ given implicitly by
$$f(x, y, z)=z^{3}+x z^{2}-y z=0, \quad f : E^{3} \rightarrow E^{1},$$
discuss the choices of $P$ and $Q$ that satisfy Theorem 4. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.
(ii) Do the same for $f(x, y, z)=e^{x y z}-1=0$.

Exercise $\PageIndex{13}$

Given $f : E^{n}\left(C^{n}\right) \rightarrow E^{m}\left(C^{m}\right), n>m,$ prove that if $f \in C D^{1}$ on a globe $G, f$ cannot be one-to-one.
[Hint for $f : E^{2} \rightarrow E^{1}:$ If, say, $D_{1} f \neq 0$ on $G, \operatorname{set} F(x, y)=(f(x, y), y)$.]

Exercise $\PageIndex{14}$

Suppose that $f$ satisfies Theorem 1 for every $\vec{p}$ in an open set $A \subseteq E^{\prime}$, and is one-to-one on $A$ (cf. Problem 4). Let $g=f_{A}^{-1}$ (restrict $f$ to $A$ and take its inverse). Show that $f$ and $g$ are open and of class $C D^{1}$ on $A$ and $f[A],$ respectively.

Exercise $\PageIndex{15}$

Given $\vec{v} \in E$ and a scalar $c \neq 0,$ define $T_{\vec{v}} : E \rightarrow E$ ("translation by $\vec{v}$") and $M_{c} : E \rightarrow E$ ("dilation by $c$"), by setting
$$T_{\vec{v}}(\vec{x})=\vec{x}+\vec{v} \text { and } M_{c}(\vec{x})=c \vec{x}.$$
Prove the following.
(i) $T_{\vec{v}}$ and $T_{\vec{v}}^{-1}\left(=T_{-\vec{v}}\right)$ are bijective, continuous, and "clopen" on $E;$ so also are $M_{c}$ and $M_{c}^{-1}\left(=M_{1 / c}\right).$
(ii) Similarly for the Lipschitz property on $E$.
(iii) If $G=G_{\vec{q}}(\delta) \subset E,$ then $T_{\vec{v}}[G]=G_{\vec{q}+\vec{v}}(\delta),$ and $M_{c}[G]=G_{c \vec{q}}(|c \delta|)$.
(iv) If $f : E^{\prime} \rightarrow E$ is linear, and $\vec{v}=f(\vec{p})$ for some $\vec{p} \in E^{\prime},$ then $T_{\vec{v}} \circ f=f \circ T_{\vec{p}}^{\prime}$ and $M_{c} \circ f=f \circ M_{c}^{\prime},$ where $T_{\vec{p}}^{\prime}$ and $M_{c}^{\prime}$ are the corresponding maps on $E^{\prime}.$ If, further, $f$ is continuous at $\vec{p},$ it is continuous on all of $E^{\prime}.$
[Hint for (iv): Fix any $\vec{x} \in E^{\prime}.$ Set $\vec{v}=f(\vec{x}-\vec{p}), g=T_{\vec{v}} \circ f \circ T_{\vec{p}-\vec{x}}^{\prime}.$ Verify that $g=f, T_{\vec{p}-\vec{x}}^{\prime}(\vec{x})=\vec{p},$ and $g$ is continuous at $\vec{x}$.]

Exercise $\PageIndex{16}$

Show that if $f : E^{\prime} \rightarrow E$ is linear and if $f\left[G^{*}\right]$ is open in $E$ for some $G^{*}=G_{\vec{p}}(\delta) \subseteq E^{\prime},$ then
(i) $f$ is open on all of $E^{\prime}$;
(ii) $f$ is onto $E$.
[Hints: (i) By Problem 8, it suffices to show that the set $f[G]$ is open, for any globe $G$ (why?). First take $G=G_{\overrightarrow{0}}(\delta).$ Then use Problems 7 and 15(i)-(iv), with suitable $\vec{v}$ and $c.$
(ii) To prove $E=f\left[E^{\prime}\right],$ fix any $\vec{y} \in E.$ As \(f=G_{\overrightarrow{0} (\delta\) is open, it contains a globe $G^{\prime}=G_{\overrightarrow{0}}(r).$ For small $c, c \vec{y} \in G^{\prime} \subseteq f\left[E^{\prime}\right].$ Hence $\vec{y} \in f\left[E^{\prime}\right]$ (Problem 10 in §2).]

Exercise $\PageIndex{17}$

Continuing Problem 16, show that if $f$ is also one-to-one on $G^{*},$ then
$$f : E^{\prime} \stackrel{\longleftrightarrow}{\text { onto }} E,$$
$f \in L\left(E^{\prime}, E\right), f^{-1} \in L\left(E, E^{\prime}\right), f$ is clopen on $E^{\prime},$ and $f^{-1}$ is so on $E$.
[Hints: To prove that $f$ is one-to-one on $E^{\prime},$ let $f(\vec{x})=\vec{y}$ for some $\vec{x}, \vec{x}^{\prime} \in E^{\prime}$. Show that
$$(\exists c, \varepsilon>0) \quad c \vec{y} \in G_{\overrightarrow{0}}(\varepsilon) \subseteq f\left[G_{\overrightarrow{0}}(\delta)\right] \text { and } f(c \vec{x}+\vec{p})=f\left(c \vec{x}^{\prime}+\vec{p}\right) \in f\left[G_{\vec{p}}(\delta)\right]=f\left[G^{*}\right].$$
Deduce that $c \vec{x}+\vec{p}=c \vec{x}^{\prime}+\vec{p}$ and $\vec{x}=\vec{x}^{\prime}.$ Then use Problem 15(v) in Chapter 4, §2, and Note 1.]

Exercise $\PageIndex{18}$

A map
$$f :(S, \rho)\stackrel{\longleftrightarrow}{\text { onto }} (T, \rho^{\prime})$$
is said to be bicontinuous, or a homeomorphism, (from $S$ onto $T)$ iff both $f$ and $f^{-1}$ are continuous. Assuming this, prove the following.
(i) $x_{n} \rightarrow p$ in $S$ iff $f\left(x_{n}\right) \rightarrow f(p)$ in $T$;
(ii) $A$ is closed (open, compact, perfect) in $S$ iff $f[A]$ is so in $T$;
(iii) $B=\overline{A}$ in $S$ iff $f[B]=\overline{f[A]}$ in $T$;
(iv) $B=A^{0}$ in $S$ iff $f[B]=(f[A])^{0}$ in $T$;
(v) $A$ is dense in $B$ (i.e., $A \subseteq B \subseteq \overline{A} \subseteq S$) in $(S, \rho)$ iff $f[A]$ is dense in $f[B] \subseteq\left(T, \rho^{\prime}\right).$
[Hint: Use Theorem 1 of Chapter 4, §2, and Theorem 4 in Chapter 3, §16, for closed sets; see also Note 1.]

Exercise $\PageIndex{19}$

Given $A, B \subseteq E, \vec{v} \in E$ and a scalar $c,$ set
$$A+\vec{v}=\{\vec{x}+\vec{v} | \vec{x} \in A\} \text { and } c A=\{c \vec{x} | \vec{x} \in A\}.$$
Assuming $c \neq 0,$ prove that
(i) $A$ is closed (open, compact, perfect) in $E$ iff $c A+\vec{v}$ is;
(ii) $B=\overline{A}$ iff $c B+\vec{v}=\overline{c A+\vec{v}}$;
(iii) $B=A^{0}$ iff $c B+\vec{v}=(c A+\vec{v})^{0}$;
(iv) $A$ is dense in $B$ iff $c A+\vec{v}$ is dense in $c B+\vec{v}$.
[Hint: Apply Problem 18 to the maps $T_{\vec{v}}$ and $M_{c}$ of Problem 15, noting that $A+\vec{v}= T_{\vec{v}}[A]$ and $c A=M_{c}[A].$]

Exercise $\PageIndex{20}$

Prove Theorem 2, for a reduced $\delta,$ assuming that only one of $E^{\prime}$ and $E$is $E^{n}\left(C^{n}\right),$ and the other is just complete.
[Hint: If, say, $E=E^{n}\left(C^{n}\right),$ then $f[\overline{G}]$ is compact (being closed and bounded), and so is $\overline{G}=f^{-1}[f[\overline{G}]].$ (Why?) Thus the Lemma works out as before, i.e., $f[G] \supseteq G_{\overline{q}}(\alpha)$.
Now use the continuity of $f$ to obtain a globe $G^{\prime}=G_{\vec{p}}\left(\delta^{\prime}\right) \subseteq G$ such that $f\left[G^{\prime}\right] \subseteq G_{\vec{q}}(\alpha).$ Let $g=f_{G}^{-1},$ further restricted to $G_{\vec{q}}(\alpha).$ Apply Problem 15(v) in Chapter 4, §2, to $g,$ with $S=G_{\vec{q}}(\alpha), T=E^{\prime}$.]