
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 $$S=G_{\vec{q}}(\alpha), T=E^{\prime}$$.]