Math 411: Honours Complex Variables - University of Alberta

26 CHAPTER 4. COMPLEX LINE INTEGRALS
Then f is locally constant, but not constant.
(i) =⇒ (iii): Let z ∈ D, and set
U := {w ∈ D : ∃ a piecewise smooth curve γ: [a,b] → D with γ(a) = z, γ(b) = w}.
Obviously, U �= ∅ (because z ∈ U).
We claim that U is open. To see this, let w0 ∈ U, so that there exists a piecewise
smooth curve γ: [a,b] → D with γ(a) = z and γ(b) = w0. Choose ǫ > 0 such that
Bǫ(w0) ⊂ D. For w ∈ Bǫ(w0), we know that [w0,w] is a curve in Bǫ(w0) ⊂ D with
initial point w0 and endpoint w. Consequently, γ ⊕ [w0,w] is a piecewise smooth
curve in D with initial point z and endpoint w. It follows that w ∈ U, and since
w ∈ Bǫ(w0) was arbitrary, we have Bǫ(w0) ⊂ U. This proves the openness of U.
Next, we claim that D \ U is also open. To see this, let w0 ∈ D \ U, and let
ǫ > 0 be so small that Bǫ(w0) ⊂ D. Assume towards a contradiction that there exists
w ∈ Bǫ(w0)∩U. Let γ: [a,b] → D be a piecewise smooth curve with γ(a) = z and
γ(b) = w. Then γ⊕[w,w0] is a piecewise smooth curve in D with initial point z and
endpoint w0, so that w0 ∈ U. This contradicts the choice of w0 ∈ D \U. It follows
that Bǫ(w0)∩U = ∅, i.e. Bǫ(w0) ⊂ D \U.
Since U and D \U are both open with U ∪(D \U) = D and U ∩(D \U) = ∅,
the connectedness of D yields that D\U = ∅, i.e. D = U.
Lemma 4.1. Suppose D ⊂ C is an open connected set (a region) and f : D → C is
continuous. Let z0 ∈ D. For each z ∈ D, let γz : [a,b] → D be a piecewise smooth
curve in D such that γz(a) = z0 and γz(b) = z. Consider the function
�
F: D → C, z ↦→ f(ζ)dζ.
For each z, let δ > 0 such that Bδ(z) ⊂ D. If the condition
�
F(w)−F(z) = f(ζ)dζ
γz
[z,w]
holds for each z and all w ∈ Bδ(z), then F is an antiderivative for f.
Proof. Let z ∈ D. Given ǫ > 0, choose δ > 0 small enough such that Bδ(z) ⊂ D and
For all w ∈ Bδ(z), we find
� �
�
�
F(w)−F(z) �
� −f(z) �
w −z � =
|ζ −z|< δ ⇒ |f(ζ)−f(z)|< ǫ.
=
��
1 �
�
|w−z| �
��
1 �
�
|w−z| �
[z,w]
[z,w]
� �
�
f − f(z) �
�
[z,w]
�
�
(f −f(z)) �
�
≤ |w−z|
|w−z| sup{|f(ζ)−f(z)|: ζ ∈ {[z,w]}}
= sup{|f(ζ)−f(z)|: ζ ∈ {[z,w]}} < ǫ.