4.8. IMPLICIT FUNCTION THEOREM 103

entries of 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

Consider fi and leth(t)≡ fi (x+ t (z−x) ,y) .

Then h(1) = h(0) and so by the mean value theorem, h′ (ti) = 0 for some ti ∈ (0,1) . There-fore, from the chain rule and for this value of ti,

h′ (ti) =n

∑j=1

∂

∂x jfi (x+ ti (z−x) ,y)(z j− x j) = 0. (4.16)

Then denote by xi the vector, x+ ti (z−x) . It follows from 4.16 that

J(x1, · · · ,xn,y

)(z−x) = 0

and so from 4.15 z−x = 0. (The matrix, in the above is invertible since its determinantis nonzero.) Now it will be shown that if η is chosen sufficiently small, then for all y ∈B(y0,η) , there exists a unique x(y) ∈ B(x0,δ ) such that f(x(y) ,y) = 0.

Claim: If η is small enough, then the function, x→ hy (x) ≡ |f(x,y)|2 achieves itsminimum value on B(x0,δ ) at a point of B(x0,δ ) . This is Proposition 4.8.5.

Choose η < η0 and also small enough that the above claim holds and let x(y) denotea point of B(x0,δ ) at which the minimum of hy on B(x0,δ ) is achieved. Since x(y) is aninterior point, you can consider hy (x(y)+ tv) for |t| small and conclude this function of thas a zero derivative at t = 0. Now

hy (x(y)+ tv) =n

∑i=1

f 2i (x(y)+ tv,y)

and so from the chain rule,

ddt

hy (x(y)+ tv) =n

∑i=1

n

∑j=1

2 fi (x(y)+ tv,y)∂ fi (x(y)+ tv,y)

∂x jv j.

Therefore, letting t = 0, it is required that for every v,

n

∑i=1

n

∑j=1

2 fi (x(y) ,y)∂ fi (x(y) ,y)

∂x jv j = 0.

In terms of matrices this reduces to 0 = 2f(x(y) ,y)T D1f(x(y) ,y)v for every vector v.Therefore, 0 = f(x(y) ,y)T D1f(x(y) ,y) . From 4.15, it follows f(x(y) ,y) = 0. Multiplyby D1f(x(y) ,y)−1 on the right. This proves the existence of the function y→ x(y) suchthat f(x(y) ,y) = 0 for all y ∈ B(y0,η) .