508 CHAPTER 24. IMPLICIT FUNCTION THEOREM*

The following is the implicit function theorem for functions of many variables. It isone of the greatest theorems in mathematics. The proof given here is like one found in oneof Caratheodory’s books on the calculus of variations and is a generalization of the abovesimpler case. I think this theorem was known to Weierstrass because it is used in a bookby Bolza which is based on his lectures. The proof follows the easier one above and isnot as elegant as some of the others which are based on a contraction mapping principlebut it may be more accessible. However, it is an advanced topic. Don’t waste your timewith it unless you have first read and understood the earlier material on linear algebra. Youwill also need the extreme value theorem for a function of n variables and the chain rule ofmulti-variable calculus. First is an interesting proposition.

Proposition 24.0.2 Suppose B(x0,δ ) ,B(y0,η0) are balls in Rn,Rm,

g : B(x0,δ )×B(y0,η0)→ [0,∞)

is continuous and g(x0,y0) = 0 and if x ̸= x0,g(x,y0) > 0. Then there exists η < η0such that if y ∈ B(y0,η) , then the function x→ g(x,y) achieves its minimum on the openset B(x0,δ ).

Proof: If not, then there is a sequence yk → y0 but the minimum of x→ g(x,yk) forx ∈ B(x0,δ ) happens on ∂B(x0,δ ) ≡ ∂B ≡ {x : |x−x0|= δ} at xk. Now ∂B is closedand bounded and so compact. Hence there is a subsequence, still denoted with subscript ksuch that xk → x ∈ ∂B and yk → y0. Let

0 < 2ε < min{g(x̂,y0) : x̂ ∈ ∂B}

Then for k large,

|g(xk,yk)−g(x,y0)|< ε, |g(xk,yk)−g(xk,y0)|< ε

the second inequality from uniform continuity. Then from these inequalities, for k large,

g(x0,yk) ≥ g(xk,yk)> g(xk,y0)− ε

> min{g(x̂,y0) : x̂ ∈ ∂B}− ε > ε

Now let k → ∞ to conclude that g(x0,y0)≥ ε , a contradiction.

Definition 24.0.3 Suppose U is an open set in Rn ×Rm and (x,y) will denote atypical point of Rn ×Rm with x ∈ Rn and y ∈ Rm. Let f : U → Rp be in C1 (U) . Thendefine

D1f (x,y) ≡

 f1,x1 (x,y) · · · f1,xn (x,y)...

...fp,x1 (x,y) · · · fp,xn (x,y)

 ,

D2f (x,y) ≡

 f1,y1 (x,y) · · · f1,ym (x,y)...

...fp,y1 (x,y) · · · fp,ym (x,y)

 .

Lemma 24.0.4 If you have an m×n matrix, M and∣∣Mi j

∣∣≤ K, for all i, j, then |Mx| ≤Km |x| where |·| is the Euclidean norm.