206 CHAPTER 8. POSITIVE LINEAR FUNCTIONALS

Lemma 8.7.12 Let f : B(p,r)→ Rn where the ball is also in Rn. Let f be one to one, fcontinuous. Then there exists δ > 0 such that

f(

B(p,r))⊇ B(f(p) ,δ ) .

In other words, f(p) is an interior point of f(

B(p,r))

.

Proof: Since f(

B(p,r))

is compact, it follows that f−1 : f(

B(p,r))→ B(p,r) is con-

tinuous. By Lemma 8.7.11, there exists a polynomial g : f(

B(p,r))→ Rn such that∥∥g− f−1

∥∥f(B(p,r)) < εr,ε < 1,

Dg(f(p))−1 exists, and g(f(p)) = f−1 (f(p)) = p

From the first inequality in the above,

|g(f(x))−x|=∣∣g(f(x))− f−1 (f(x))

∣∣≤ ∥∥g− f−1∥∥f(B(p,r)) < εr

By Lemma 8.7.10,

g◦ f(

B(p,r))⊇ B(p,(1− ε)r) = B(g(f(p)) ,(1− ε)r)

Since Dg(f(p))−1 exists, it follows from the inverse function theorem that g−1 also existsand that g,g−1 are open maps on small open sets containing f(p) and p respectively. Thusthere exists η < (1− ε)r such that g−1 is an open map on B(p,η)⊆ B(p,(1− ε)r). Thus

g◦ f(

B(p,r))⊇ B(p,(1− ε)r)⊇ B(p,η)

So do g−1‘ to both ends. Then you have g−1 (p) = f(p) is in the open set g−1 (B(p,η)) .Thus

f(

B(p,r))⊇ g−1 (B(p,η))⊇ B

(g−1 (p) ,δ

)= B(f(p) ,δ ) ■

pq◦ f

(B(p,r)

)B(p,(1− ε)r))

p = q(f(p))

With this lemma, the invariance of domain theorem comes right away. This remark-able theorem states that if f : U → Rn for U an open set in Rn and if f is one to one andcontinuous, then f(U) is also an open set in Rn.

Theorem 8.7.13 Let U be an open set in Rn and let f : U → Rn be one to one andcontinuous. Then f(U) is also an open subset in Rn.

Proof: It suffices to show that if p ∈ U then f(p) is an interior point of f(U). LetB(p,r) ⊆ U. By Lemma 8.7.12, f(U) ⊇ f

(B(p,r)

)⊇ B(f(p) ,δ ) so f(p) is indeed an

interior point of f(U). ■