
“real: chapter_12” — 2011/5/23 — 1:04 — page 20 — #20
12-20 Real Analysis
Differentiability: Let x ∈[a −h, a +h]. Choose x +p ∈[a −h, a +h].
Let y = φ(x), y +q = φ(x +p) so that f (x, y) = 0, f (x +p, y +q) = 0.
f (x + p, y + q) − f (x, y) = f (x + p, y + q) − f (x + p, y)
+ f (x + p, y) − f (x, y)
= qf
y
(x + p, y + qθ
1
) + pf
x
(x + pθ
2
,y)
(by Theorem 12.3.10)
with 0 <θ
1
, θ
2
< 1. f (x + p, y + q) − f (x, y) = 0 implies that
qf
y
(x + p, y + qθ
1
) + pf
x
(x + pθ
2
, y) = 0.
Since f
y
(x, y) = 0 for (x, y ) ∈ R and (x + p, y + qθ
1
) ∈ R, we have
φ(x + p) − φ(x)
p
=
q
p
=−
f
x
(x + pθ
2
, y)
f
y
(x + p, y + qθ
1
)
.
Since φ is continuous, q → 0asp → 0. Therefore, f
x
and f
y
being
continuous, ...