Page 49 - Vector Analysis

§2.4 Conditions for Differentiability 45

Proof. W.L.O.G., we assume that f is a function of two variables; that is, n = 2. For fixed
h, k P R, define φ(x, y) = f (x, y + k) ´ f (x, y) and ψ(x, y) = f (x + h, y) ´ f (x, y). Then

φ(a + h, b) ´ φ(a, b) = f (a + h, b + k) ´ f (a + h, b) ´ f (a, b + k) + f (a, b)
= ψ(a, b + k) ´ ψ(a, b) .

By the mean value theorem (Theorem A.9), for h, k ‰ 0 and sufficiently small,
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φ(a + h, b) ´ φ(a, b) = φx(a + θ1h, b)h = fx(a + θ1h, b + k) ´ fx(a + θ1h, b) h
= (fx)y(a + θ1h, b + θ2k)hk

for some θ1, θ2 P (0, 1), and similarly, for some θ3, θ4 P (0, 1),

ψ(a, b + k) ´ ψ(a, b) = (fy)x(a + θ3h, b + θ4k)hk .

Therefore, for h, k ‰ 0 and sufficiently small, there exist θ1, θ2, θ3, θ4 P (0, 1) such that

(fx)y(a + θ1h, b + θ2k) = (fy)x(a + θ3h, b + θ4k) . (2.1)

Let ε ą 0 be given. Since (fx)y and (fy)x are continuous at (a, b), there exist δ1, δ2 ą 0
such that

ˇˇ(fx)y(x, y) ´ (fx)y(a, b)ˇˇ ă ε if a(x ´ a)2 + (y ´ b)2 ă δ1 ,
2
ε
ˇˇ(fx)y(x, y) ´ (fx)y(a, b)ˇˇ ă 2 if a(x ´ a)2 + (y ´ b)2 ă δ2 .

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In particular, if δ = mintδ1, δ2u and h, k ‰ 0 satisfying h2 + k2 ă δ,

ˇˇ(fx)y(a + θ1h, b + θ2k) ´ (fx)y(a, b)ˇˇ + ˇˇ(fx)y(a + θ3h, b + θ4k) ´ (fx)y(a, b)ˇˇ ă ε ,

where θ1, θ2, θ3, θ4 P (0, 1) are chosen to validate (2.1). As a consequence,

ˇˇ(fx)y(a, b) ´ (fy)x(a, b)ˇˇ
= ˇˇ(fx)y(a, b) ´ (fx)y(a + θ1h, b + θ2k) + (fx)y(a + θ3h, b + θ4k) ´ (fx)y(a, b)ˇˇ
ď ˇˇ(fx)y(a + θ1h, b + θ2k) ´ (fx)y(a, b)ˇˇ + ˇˇ(fx)y(a + θ3h, b + θ4k) ´ (fx)y(a, b)ˇˇ ă ε

which concludes the theorem (since ε ą 0 is given arbitrarily). ˝