1620 CHAPTER 51. FUNDAMENTALS OF COMPLEX ANALYSIS

Proof: Let Λ ∈ X ′. Then Λg : [a,b]→ C . Therefore, from Lemma 51.3.2, for eachΛ ∈ X ′,Λg(s) = Λg(t) and since X ′ separates the points, it follows g(s) = g(t) so g isconstant.

Lemma 51.3.5 Let φ : [a,b]× [c,d]→ R be continuous and let

g(t)≡∫ b

aφ (s, t)ds. (51.3.1)

Then g is continuous. If ∂φ

∂ t exists and is continuous on [a,b]× [c,d] , then

g′ (t) =∫ b

a

∂φ (s, t)∂ t

ds. (51.3.2)

Proof: The first claim follows from the uniform continuity of φ on [a,b]× [c,d] , whichuniform continuity results from the set being compact. To establish 51.3.2, let t and t + hbe contained in [c,d] and form, using the mean value theorem,

g(t +h)−g(t)h

=1h

∫ b

a[φ (s, t +h)−φ (s, t)]ds

=1h

∫ b

a

∂φ (s, t +θh)∂ t

hds

=∫ b

a

∂φ (s, t +θh)∂ t

ds,

where θ may depend on s but is some number between 0 and 1. Then by the uniformcontinuity of ∂φ

∂ t , it follows that 51.3.2 holds.

Corollary 51.3.6 Let φ : [a,b]× [c,d]→ C be continuous and let

g(t)≡∫ b

aφ (s, t)ds. (51.3.3)

Then g is continuous. If ∂φ

∂ t exists and is continuous on [a,b]× [c,d] , then

g′ (t) =∫ b

a

∂φ (s, t)∂ t

ds. (51.3.4)

Proof: Apply Lemma 51.3.5 to the real and imaginary parts of φ .Applying the above corollary to the components, you can also have the same result for

φ having values in Cn.

Corollary 51.3.7 Let φ : [a,b]× [c,d]→ Cn be continuous and let

g(t)≡∫ b

aφ (s, t)ds. (51.3.5)

Then g is continuous. If ∂φ

∂ t exists and is continuous on [a,b]× [c,d] , then

g′ (t) =∫ b

a

∂φ (s, t)∂ t

ds. (51.3.6)