509

Proof: This follows from a computation.

|Mx| =

∑i

∣∣∣∣∣∑jMi jx j

∣∣∣∣∣21/2

≤

∑i

(∑

j

∣∣Mi jx j∣∣)21/2

≤ K ∑i

(∑

j

∣∣x j∣∣2)1/2

= Km |x|

I will use this observation whenever convenient in the following proof.

Theorem 24.0.5 (implicit function theorem) Suppose U is an open set in Rn ×Rm.Let f : U → Rn be in C1 (U) and suppose

f (x0,y0) = 0, D1f (x0,y0)−1 exists. (24.1)

Then there exist positive constants, δ ,η , such that for every y ∈ B(y0,η) there exists aunique x(y) ∈ B(x0,δ ) such that

f (x(y) ,y) = 0. (24.2)

Furthermore, the mapping, y → x(y) is in C1 (B(y0,η)).

Proof: Let

f (x,y) =(

f1 (x,y) f2 (x,y) · · · fn (x,y))T

.

Define for(x1, · · · ,xn

)∈ B(x0,δ )

nand y ∈ B(y0,η) the following matrix.

J(x1, · · · ,xn,y

)≡

 f1,x1

(x1,y

)· · · f1,xn

(x1,y

)...

...fn,x1 (x

n,y) · · · fn,xn (xn,y)

 . (*)

Then by the assumption of continuity of all the partial derivatives, there exists r > 0 andδ 0,η0 > 0 such that if δ ≤ δ 0 and η ≤η0, it follows that for all

(x1, · · · ,xn

)∈B(x0,δ )

n ≡B(x0,δ )×B(x0,δ )×·· ·×B(x0,δ ), and y ∈ B(y0,η),∣∣det

(J(x1, · · · ,xn,y

))∣∣> r > 0. (24.3)

and B(x0,δ 0)× B(y0,η0)⊆U . Therefore, from the formula for the inverse of a matrix andcontinuity of all entries of the various matrices, there exists a constant K such that all entriesof J

(x1, · · · ,xn,y

),J(x1, · · · ,xn,y

)−1, and D2f (x,y) have absolute value smaller than

K on the convex set B(x0,δ )n×B(y0,η) whenever δ ,η are sufficiently small. It is always

tacitly assumed that these radii are this small.Next it is shown that for a given y ∈ B(y0,η) ,η ≤ η0, there is at most one x ∈

B(x0,δ 0) such that f (x,y) = 0.Pick y ∈ B(y0,η) and suppose there exist x,z ∈ B(x0,δ ) such that

f (x,y) = f (z,y) = 0