116 CHAPTER 6. MULTI-VARIABLE CALCULUS

Proof: Suppose first the second condition holds. Then from the material on lineartransformations, ∣∣Ai j (x)−Ai j (y)

∣∣ =∣∣ei · (A(x)−A(y))e j

∣∣≤ |ei|

∣∣(A(x)−A(y))e j∣∣

≤ ||A(x)−A(y)|| .

Therefore, the second condition implies the first.Now suppose the first condition holds. That is each Ai j is continuous at x. Let |v| ≤ 1.

|(A(x)−A(y))(v)| =

∑i

∣∣∣∣∣∑j(Ai j (x)−Ai j (y))v j

∣∣∣∣∣21/2

(6.11.13)

≤

∑i

(∑

j

∣∣Ai j (x)−Ai j (y)∣∣ ∣∣v j

∣∣)21/2

.

By continuity of each Ai j, there exists a δ > 0 such that for each i, j∣∣Ai j (x)−Ai j (y)∣∣< ε

n√

m

whenever |x−y|< δ . Then from 6.11.13, if |x−y|< δ ,

|(A(x)−A(y))(v)| <

∑i

(∑

j

ε

n√

m|v|)21/2

≤

∑i

(∑

j

ε

n√

m

)21/2

= ε

This proves the proposition.

6.12 The Frechet DerivativeLet U be an open set in Fn, and let f : U → Fm be a function.

Definition 6.12.1 A function g is o(v) if

lim|v|→0

g(v)|v|

= 0 (6.12.14)

A function f : U → Fm is differentiable at x ∈ U if there exists a linear transformationL ∈L (Fn,Fm) such that

f(x+v) = f(x)+Lv+o(v)

This linear transformation L is the definition of Df(x). This derivative is often called theFrechet derivative. .