As an application of the inverse function theorem is a simple proof of the important invariance of domain
theorem which says that continuous and one to one functions defined on an open set in ℝ^{n} with values in
ℝ^{n} take open sets to open sets. You know that this is true for functions of one variable because a one to
one continuous function must be either strictly increasing or strictly decreasing. This will be used when
considering orientations of curves later. However, the n dimensional version isn’t at all obvious but is just
as important if you want to consider manifolds with boundary for example. The need for this
theorem occurs in many other places as well in addition to being extremely interesting for its own
sake.
Lemma 3.7.1Let f be continuous and map B
(p,r)
⊆ ℝ^{n}to ℝ^{n}. Suppose that for all x ∈B
(p,r)
,
|f (x) − x | < εr
Then it follows that
(------)
f B (p,r) ⊇ B (p,(1 − ε)r)
Proof:This is from the Brouwer fixed point theorem, Corollary 2.10.4. Consider for
y ∈ B
(p,(1 − ε)r)
h (x) ≡ x − f (x )+ y
Then h is continuous and for x ∈B
(p,r)
,
|h(x)− p| = |x− f (x) +y − p| < εr+ |y − p | < εr+ (1− ε)r = r
Hence h :B
(p,r)
→B
(p,r)
and so it has a fixed point x by Corollary 2.10.4. Thus
x− f (x) +y = x
so f
(x)
= y. ■
The notation
∥f∥
_{K} means sup_{x∈K}
|f (x )|
.
Lemma 3.7.2Let K be a compact set in ℝ^{n}and let g : K → ℝ^{n}be continuous,z ∈ K is fixed. Let δ > 0.Then there exists a polynomial q (each component a polynomial) such that
∥q − g ∥K < δ, q(z) = g (z), Dq (z)−1 exists
Proof:By the Weierstrass approximation theorem, Theorem 2.8.11, (apply this theorem to the algebra
of real polynomials) there exists a polynomial
(------)
q∘ f B (p,r) ⊇ B (p, (1 − ε) r) = B (q (f (p )),(1− ε)r)
So do q^{−1‘} to both sides. Then you would have f
( )
B-(p,r)
⊇ q^{−1}
(B (q(f (p)),(1− ε)r))
, and it becomes
a question whether the set on the right contains an open set B
(f (p ),δ)
and whether g^{−1}
exists. This is where the inverse function theorem is used. One reduces
(1 − ε)
r to make this
happen.
PICT
By the inverse function theorem, there is an open set containing f
(p)
denoted as W such that on W, q
is one to one and q and its inverse, defined on an open set V = q
(W )
both map open sets to open
sets.
PICT
By the construction, p = q
(f (p))
∈ V and so if η is small enough, it follows that
(------)
B (p,η) = B (q(f (p)),η) ⊆ B (p,(1 − ε)r)∩V ⊆ q ∘f B (p,r) .
and q,q^{−1} both map open sets to open sets. Thus q^{−1}
(B (q (f (p)) ,η))
is an open set containing f
(p)
and
this open set is contained in f
(B-(p,r))
. Hence if δ is small enough,
( )
f B-(p,r) ⊇ B (f (p),δ) ■
With this lemma, the invariance of domain theorem comes right away. This remarkable theorem states
that if f : U → ℝ^{n} for U an open set in ℝ^{n} and if f is one to one and continuous, then f
(U )
is also an open
set in ℝ^{n}.
Theorem 3.7.4Let U be an open set in ℝ^{n}and let f : U → ℝ^{n}be oneto one and continuous.Then f