762 CHAPTER 23. DEGREE THEORY

last equations hold in what follows,

d (f,Ω,y) = ∑{

sgn(det(Dg(x))) : x ∈ g−1 (y)}

=∫

Ω

φ ε (h(x,1))detD1h(x,1)dx

=∫

Ω

φ ε (h(x,0))detD1h(x,0)dx

= ∑{

sgn(det(Dĝ(x))) : x ∈ g−1 (y)}

The last claim was noted earlier. If y /∈ f(Ω), then letting g be smooth with ∥f−g∥

∞,Ω <

δ < dist(f(∂Ω) ,y) , it follows that if x ∈ ∂Ω,

∥g(x)−y∥ ≥ ∥y− f(x)∥−∥g(x)− f(x)∥> dist(f(∂Ω) ,y)−δ ≥ 0

Thus, from the definition, d (f,Ω,y) = 0.

23.2 Properties of the DegreeNow that the degree for a continuous function has been defined, it is time to considerproperties of the degree. In particular, it is desirable to prove a theorem about homotopyinvariance which depends only on continuity considerations.

Theorem 23.2.1 If h is in C(Ω× [a,b] ,Rp

), and 0 /∈ h(∂Ω× [a,b]) then

t→ d (h(·, t) ,Ω,0)

is constant for t ∈ [a,b].

Proof: Let 0 < δ < mint∈[a,b] dist(h(∂Ω× [a,b]) ,0) . By Corollary 23.1.1, there exists

hm (·, t) =m

∑k=0

pk (t)h(·, tk)

for pk (t) some polynomial in t of degree m such that

maxt∈[a,b]

∥hm (·, t)−h(·, t)∥∞,Ω < δ (23.2.5)

Letting ψn be a mollifier,

C∞c

(B(

0,1n

)),∫Rp

ψn (u)du = 1

letgn (·, t)≡ hm ∗ψn (·, t)

Thus,

gn (x, t) ≡∫Rp

hm (x−u, t)ψn (u)du =m

∑k=0

pk (t)∫Rp

h(x−u, tk)ψn (u)du

=m

∑k=0

pk (t)∫Rp

h(u, tk)ψn (x−u)du≡m

∑k=0

pk (t)h(·, tk)∗ψn (x)(23.2.6)