488 CHAPTER 26. IMPLICIT FUNCTION THEOREM*

where yi is a point on the line segment joining y1 and y2. Thus from 26.4 and the Cauchy-Schwarz inequality,

∣∣D2 fi(x(y2) ,y

i)(y1−y2)

∣∣ ≤ K |y1−y2| . Therefore, defining thesymbol M

(y1, · · · ,yn

)≡M denote the matrix having the ith row equal to

D2 fi(x(y2) ,y

i) ,it follows

|M (y1−y2)| ≤

(∑

iK2 |y1−y2|

2

)1/2

=√

mK |y1−y2| . (26.8)

Also, from 26.7,

J(x1, · · · ,xn,y1

)(x(y1)−x(y2)) =−M (y1−y2) (26.9)

and so from 26.8, 26.5, |x(y1)−x(y2)|=

=∣∣∣J (x1, · · · ,xn,y1

)−1M (y1−y2)

∣∣∣=

(n

∑i=1

∣∣∣eTi J(x1, · · · ,xn,y1

)−1M (y1−y2)

∣∣∣2)1/2

≤

(n

∑i=1

K2 |M (y1−y2)|2

)1/2

≤

(n

∑i=1

K2 (√mK |y1−y2|)2

)1/2

= K2√mn |y1−y2|

Now let y2 = y,y1 = y+hek for small h. Then M depends on h and

limh→0

M (h) = D2f (x(y) ,y)

thanks to the continuity of y→ x(y) just shown. Also,

x(y+hek)−x(y)

h=−J

(x1 (h) , · · · ,xn (h) ,y+hek

)−1M (h)ek

Passing to a limit and using the formula for the inverse of a matrix in terms of the cofactormatrix, and the continuity of y→ x(y) shown above, this yields

∂x

∂yk=−D1f (x(y) ,y)−1 D2 fi (x(y) ,y)ek

Then continuity of y→ x(y) and the assumed continuity of the partial derivatives of fshows that each partial derivative of y→ x(y) exists and is continuous. ■

This implies the inverse function theorem given next.

Theorem 26.0.3 (inverse function theorem) Let x0 ∈ U, an open set in Rn , and let f :U → Rn. Suppose

f is C1 (U) , and Df(x0)−1 exists. (26.10)

Then there exist open sets W, and V such that

x0 ∈W ⊆U, (26.11)

f : W →V is one to one and onto, (26.12)

f−1 is C1, (26.13)