512 CHAPTER 24. IMPLICIT FUNCTION THEOREM*

Theorem 24.1.1 (implicit function theorem) Suppose U is an open set in Fn ×Fm.Let f : U → Fn be in Ck (U) and suppose

f (x0,y0) = 0, D1f (x0,y0)−1 ∈ L (Fn,Fn) . (24.11)

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. (24.12)

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

Proof: From the implicit function theorem y → x(y) is C1. It remains to show that itis Ck for k > 1 assuming that f is Ck. From 24.12

∂x

∂yl =−D1f (x,y)−1 ∂f

∂yl .

Thus the following formula holds for q = 1 and |α|= q.

Dαx(y) = ∑|β |≤q

Mβ (x,y)Dβf (x,y) (24.13)

where Mβ is a matrix whose entries are differentiable functions of Dγx for |γ| < q andDτf (x,y) for |τ| ≤ q. This follows easily from the description of D1f (x,y)−1 in terms ofthe cofactor matrix and the determinant of D1f (x,y). Suppose 24.13 holds for |α|= q< k.Then by induction, this yields x is Cq. Then

∂Dαx(y)

∂yp = ∑|β |≤|α|

∂Mβ (x,y)

∂yp Dβf (x,y)+Mβ (x,y)∂Dβf (x,y)

∂yp .

By the chain rule∂Mβ (x,y)

∂yp is a matrix whose entries are differentiable functions of

Dτf (x,y)

for |τ| ≤ q+ 1 and Dγx for |γ| < q+ 1. It follows, since yp was arbitrary, that for any|α|= q+1, a formula like 24.13 holds with q being replaced by q+1. By induction, x isCk.

As a simple corollary, this yields the inverse function theorem. You just let F (x,y) =y−f (x) and apply the implicit function theorem.

Theorem 24.1.2 (inverse function theorem) Let x0 ∈ U ⊆ Fn and let f : U → Fn.Suppose for k a positive integer,

f is Ck (U) , and Df(x0)−1 ∈ L (Fn,Fn). (24.14)

Then there exist open sets W, and V such that

x0 ∈W ⊆U, (24.15)

f : W →V is one to one and onto, (24.16)

f−1 is Ck. (24.17)