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Mathematics LibreTexts

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

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Exercise 6.7.E.1

Discuss: In Definition 1, ¯G can equivalently be replaced by G=Gp(δ) (an open globe).

Exercise 6.7.E.2

Prove that if the set D is open (closed) in (S,ρ), then the map f:ST is open (closed, respectively) on D iff fD(f restricted to D) has this property as a map of D into f[D].
[Hint: Use Theorem 4 in Chapter 3, §12.]

Exercise 6.7.E.3

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

Exercise 6.7.E.3

Verify footnotes 2 and 3.

Exercise 6.7.E.4

Show that a map f:EE may fail to be one-to-one on all of E even if f satisfies Theorem 1 near every pE. Nonetheless, show that this cannot occur if E=E=E1.
[Hints: For the first part, take E=C,f(x+iy)=ex(cosy+isiny). For the second, use Theorem 1 in Chapter 5, §2.]

Exercise 6.7.E.4

(i) For maps f:E1E1, prove that the existence of a bijective df(p;) is equivalent to f(p)0.
(ii) Let
f(x)=x+x2sin1x,f(0)=0.
Show that f(0)0, and fCD1 near any p0; yet f is not one-to-one near 0. What is wrong?

Exercise 6.7.E.5

Show that a map f:En(Cn)En(Cn),fCD1, may be bijective even if det[f(p)]=0 at some p, but then f1 cannot be differentiable at q=f(p).
[Hint: For the first clause, take f(x)=x3,p=0; for the second, note that if f1 is differentiable at q, then Note 2 in §4 implies that det [df(p;)]det[df1(q;)]=10, since ff1 is the identity map.]

Exercise 6.7.E.6

Prove Corollary 2 for the general case of complete E and E.
[Outline: Given a closed X¯G, take any convergent sequence {yn}f[X]. By Problem 8 in Chapter 4, §8, f1(yn)=xn is a Cauchy sequence in X (why?). By the completeness of E,(xX)xnx (Theorem 4 of Chapter 3, §16). Infer that limyn=f(x)f[X], so f[X] is closed.]

Exercise 6.7.E.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 6.7.E.8

Prove in detail that f:(S,ρ)(T,ρ) is open on DS iff f maps the interior of D into that of f[D]; that is, f[D0](f[D])0.

Exercise 6.7.E.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:E2E1 by f(x,y)=x and let
D={(x,y)E2|y=1x,x>0};
use Theorem 4(iii) in Chapter 3, §16 and continuity to show that D is closed in E2, but f[D]=(0,+) is not closed in E1. However, f is open on all of E2 by Problem 8. (Verify!)]

Exercise 6.7.E.10

Continuing Problem 9(ii), define f:EnE1 (or CnC) by f(x)= xk for a fixed kn (the "kth projection map"). Show that f is open, but not closed, on En(Cn).

Exercise 6.7.E.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 6.7.E.12

(i) Treating z as a function of x and y, given implicitly by
f(x,y,z)=z3+xz2yz=0,f:E3E1,
discuss the choices of P and Q that satisfy Theorem 4. Find zx and zy.
(ii) Do the same for f(x,y,z)=exyz1=0.

Exercise 6.7.E.13

Given f:En(Cn)Em(Cm),n>m, prove that if fCD1 on a globe G,f cannot be one-to-one.
[Hint for f:E2E1: If, say, D1f0 on G,setF(x,y)=(f(x,y),y).]

Exercise 6.7.E.14

Suppose that f satisfies Theorem 1 for every p in an open set AE, and is one-to-one on A (cf. Problem 4). Let g=f1A (restrict f to A and take its inverse). Show that f and g are open and of class CD1 on A and f[A], respectively.

Exercise 6.7.E.15

Given vE and a scalar c0, define Tv:EE ("translation by v") and Mc:EE ("dilation by c"), by setting
Tv(x)=x+v and Mc(x)=cx.
Prove the following.
(i) Tv and T1v(=Tv) are bijective, continuous, and "clopen" on E; so also are Mc and M1c(=M1/c).
(ii) Similarly for the Lipschitz property on E.
(iii) If G=Gq(δ)E, then Tv[G]=Gq+v(δ), and Mc[G]=Gcq(|cδ|).
(iv) If f:EE is linear, and v=f(p) for some pE, then Tvf=fTp and Mcf=fMc, where Tp and Mc are the corresponding maps on E. If, further, f is continuous at p, it is continuous on all of E.
[Hint for (iv): Fix any xE. Set v=f(xp),g=TvfTpx. Verify that g=f,Tpx(x)=p, and g is continuous at x.]

Exercise 6.7.E.16

Show that if f:EE is linear and if f[G] is open in E for some G=Gp(δ)E, then
(i) f is open on all of E;
(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=G0(δ). Then use Problems 7 and 15(i)-(iv), with suitable v and c.
(ii) To prove E=f[E], fix any yE. As \(f=G_{\overrightarrow{0} (\delta\) is open, it contains a globe G=G0(r). For small c,cyGf[E]. Hence yf[E] (Problem 10 in §2).]

Exercise 6.7.E.17

Continuing Problem 16, show that if f is also one-to-one on G, then
f:E onto E,
fL(E,E),f1L(E,E),f is clopen on E, and f1 is so on E.
[Hints: To prove that f is one-to-one on E, let f(x)=y for some x,xE. Show that
(c,ε>0)cyG0(ε)f[G0(δ)] and f(cx+p)=f(cx+p)f[Gp(δ)]=f[G].
Deduce that cx+p=cx+p and x=x. Then use Problem 15(v) in Chapter 4, §2, and Note 1.]

Exercise 6.7.E.18

A map
f:(S,ρ) onto (T,ρ)
is said to be bicontinuous, or a homeomorphism, (from S onto T) iff both f and f1 are continuous. Assuming this, prove the following.
(i) xnp in S iff f(xn)f(p) in T;
(ii) A is closed (open, compact, perfect) in S iff f[A] is so in T;
(iii) B=¯A in S iff f[B]=¯f[A] in T;
(iv) B=A0 in S iff f[B]=(f[A])0 in T;
(v) A is dense in B (i.e., AB¯AS) in (S,ρ) iff f[A] is dense in f[B](T,ρ).
[Hint: Use Theorem 1 of Chapter 4, §2, and Theorem 4 in Chapter 3, §16, for closed sets; see also Note 1.]

Exercise 6.7.E.19

Given A,BE,vE and a scalar c, set
A+v={x+v|xA} and cA={cx|xA}.
Assuming c0, prove that
(i) A is closed (open, compact, perfect) in E iff cA+v is;
(ii) B=¯A iff cB+v=¯cA+v;
(iii) B=A0 iff cB+v=(cA+v)0;
(iv) A is dense in B iff cA+v is dense in cB+v.
[Hint: Apply Problem 18 to the maps Tv and Mc of Problem 15, noting that A+v=Tv[A] and cA=Mc[A].]

Exercise 6.7.E.20

Prove Theorem 2, for a reduced δ, assuming that only one of E and Eis En(Cn), and the other is just complete.
[Hint: If, say, E=En(Cn), then f[¯G] is compact (being closed and bounded), and so is ¯G=f1[f[¯G]]. (Why?) Thus the Lemma works out as before, i.e., f[G]G¯q(α).
Now use the continuity of f to obtain a globe G=Gp(δ)G such that f[G]Gq(α). Let g=f1G, further restricted to Gq(α). Apply Problem 15(v) in Chapter 4, §2, to g, with S=Gq(α),T=E.]


6.7.E: Problems on Inverse and Implicit Functions, Open and Closed Maps is shared under a CC BY 1.0 license and was authored, remixed, and/or curated by LibreTexts.

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