Page 46 - Vector Analysis

42 CHAPTER 2. Differentiation of Functions of Several Variables

Theorem 2.30. Let U Ď Rn be open, a P U , and f : U Ñ R. If
1. the Jacobian matrix of f exists in a neighborhood of a, and

2. at least (n ´ 1) entries of the Jacobian matrix of f are continuous at a,

then f is differentiable at a.

Proof. W.L.O.G. we can assume that Bf , Bf , ¨ ¨ ¨ , Bf are continuous at a. Let tejunj=1
B x2 B xn´1
B x1
be the standard basis of Rn, and ε ą 0 be given. Since B f is continuous at a for i =
B xi
1, ¨ ¨ ¨ , n ´ 1,

D δi ą 0 Q ˇ Bf (x) ´ Bf ˇ ă ε whenever }x ´ a}Rn ă δi .
ˇ B xi (a)ˇ ?
ˇ B xi ˇ
n

On the other hand, by the definition of the partial derivatives,

D δn ą 0 Q ˇ f (a + hen) ´ f (a) ´ Bf ˇ ă ε whenever 0 ă |h| ă δn .
ˇ (a)ˇ ?

ˇh B xn ˇ n

Let k = x ´ a and δ = min ␣δ1, ¨ ¨ ¨ , δn(. Then

ˇ [ Bf Bf ]ˇ
ˇf (x) B xn
ˇ ´ f (a) ´ B x1 (a)(x1 ´ a1) + ¨ ¨ ¨ + (a)(xn ´ an) ˇ
ˇ

= ˇ + k) ´ f (a) ´ Bf (a)k1 ´ ¨ ¨ ¨ ´ Bf (a)kn ˇ
ˇf (a B x1 B xn ˇ
ˇ ˇ

ˇ Bf Bf ˇ
B x1 B xn ˇ
= ˇf (a1 + k1, ¨ ¨ ¨ , an + kn) ´ f (a1, ¨ ¨ ¨ , an) ´ (a)k1 ´ ¨ ¨ ¨ ´ (a)kn ˇ
ˇ

ˇ Bf ˇ
B x1 (a)k1ˇˇ
ď ˇf (a1 + k1, ¨ ¨ ¨ , an + kn) ´ f (a1, a2 + k2, ¨ ¨ ¨ , an + kn) ´
ˇ

ˇ Bf ˇ
B x2 (a)k2ˇˇ
+ ˇf (a1, a2 + k2, ¨ ¨ ¨ , an + kn) ´ f (a1, a2, a3 + k3, ¨ ¨ ¨ , an + kn) ´
ˇ

ˇ Bf ˇ
B xn (a)knˇˇ
+ ¨ ¨ ¨ + ˇf (a1 , ¨ ¨ ¨ , an´1, an + kn) ´ f (a1, ¨ ¨ ¨ , an) ´ .
ˇ

By the mean value theorem,

f (a1, ¨ ¨ ¨ , aj´1, aj + kj, ¨ ¨ ¨ , an + kn) ´ f (a1, ¨ ¨ ¨ , aj, aj+1 + kj+1, ¨ ¨ ¨ , an + kn)

= kj Bf (a1, ¨ ¨ ¨ , aj´1, aj + θj kj , aj+1 + kj+1, ¨ ¨ ¨ , an + kn)
B xj

for some 0 ă θj ă 1; thus for j = 1, ¨ ¨ ¨ , n ´ 1, if }x ´ a}Rn = }k}Rn ă δ,

ˇ Bf ˇ
B xj
ˇf (a1, ¨ ¨ ¨ , aj´1, aj + kj , ¨ ¨ ¨ , an + kn) ´ f (a1, ¨¨ ¨ , aj , aj+1 + kj+1, ¨ ¨ ¨ , an + kn) ´ (a)kj ˇ
ˇ ˇ

= ˇ Bf (a1, , aj´1, aj + θjkj, aj+1 + kj+1, ¨ ¨ ¨ , an + kn) Bf ˇ ε |kj .
ˇ B xj ¨ ¨ ¨ ´ B xj (a)ˇˇ|kj | ď ? |
ˇ n