4.9. MORE CONTINUOUS PARTIAL DERIVATIVES 105

4.9 More Continuous Partial DerivativesThe implicit function theorem will now be improved slightly. If f is Ck, it follows thatthe function which is implicitly defined is also Ck, not just C1, meaning all mixed partialderivatives of f up to order k are continuous. Since the inverse function theorem comes as acase of the implicit function theorem, this shows that the inverse function also inherits theproperty of being Ck. First some notation is convenient. Let α = (α1, · · · ,αn) where eachα i is a nonnegative integer. Then letting |α|= ∑i α i,

Dα f(x)≡ ∂ |α|f∂ α1∂ α2 · · ·∂ αn

(x) , D0f(x)≡ f(x)

The symbol on the right means to take the αn partial derivative with respect to xn, then theαn−1 partial derivative with respect to xn−1 of what you just got and so on till you take theα1 partial derivative with respect to x1. The idea is to show that all mixed partial derivativessuch that |α| ≤ k exist and are continuous.

Theorem 4.9.1 (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) . (4.19)

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. (4.20)

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 it isCk for k > 1 assuming that f is Ck. From 4.20

∂x∂yl =−D1f(x,y)−1 ∂ f

∂yl .

By the formula for the inverse in terms of cofactors, if f is C2, one can use the chain ruleto take another continuous derivative. Thus, the following formula holds for q = 1 and|α|= q.

Dα x(y) = ∑|β |≤q

Mβ (x,y)Dβ f(x,y) (4.21)

where Mβ is a matrix whose entries are differentiable functions of Dγ x for |γ| < q andDτ f(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 4.21 holds for |α|= q < k.Then by induction, this yields x is Cq. Then

∂Dα x(y)∂yp = ∑

|β |≤|α|

∂Mβ (x,y)∂yp Dβ f(x,y)+Mβ (x,y)

∂Dβ f(x,y)∂yp .

By the chain rule∂Mβ (x,y)

∂yp is a matrix whose entries are differentiable functions of Dτ f(x,y)for |τ| ≤ q+ 1 and Dγ x for |γ| < q+ 1. It follows, since yp was arbitrary, that for any