194 CHAPTER 7. THE DERIVATIVE

7.8 Some Standard NotationIn the case where X = Rn there is a special notation which is often used to describe higherorder mixed partial derivatives. It is called multi-index notation.

Definition 7.8.1 α = (α1, · · · ,αn) for α1 · · ·αn positive integers is called a multi-index, as before with polynomials. For α a multi-index, |α| ≡ α1 + · · ·+αn, and if x ∈ X,

x= (x1, · · · ,xn),

and f a function, define

xα ≡ xα11 xα2

2 · · ·xαnn , Dαf(x)≡ ∂ |α|f(x)

∂xα11 ∂xα2

2 · · ·∂xαnn

.

Then in this special case, the following is another description of what is meant by a Ck

function.

Definition 7.8.2 Let U be an open subset of Rn and let f : U → Y. Then for k anonnegative integer, a differentiable function f is Ck if for every |α| ≤ k, Dαf exists andis continuous.

Theorem 7.8.3 Let U be an open subset of Rn and let f : U → Y. Then if Drf (x)exists for r ≤ k, then Drf is continuous at x for r ≤ k if and only if Dαf is continuous at xfor each |α| ≤ k.

Proof: First consider the case of a single derivative. Then as shown above, the matrixof Df (x) is just

J (x)≡(

∂f∂x1

(x) · · · ∂f∂xn

(x))

and to say that x→ Df (x) is continuous is the same as saying that each of these partialderivatives is continuous. Written out in more detail,

f (x+v)−f (x) = Df (x)v+o(v) =n

∑k=1

∂f

∂xk(x)vk +o(v)

Thus Df (x)v = ∑nk=1

∂f∂xk

(x)vk. Now consider the second derivative.

D2f (x)(w)(v) =

Df (x+w)v−Df (x)v+o(w)(v)=n

∑k=1

(∂f

∂xk(x+w)− ∂f

∂xk(x)

)vk +o(w)(v)

=n

∑k=1

(n

∑j=1

∂ 2f (x)

∂x j∂xkw j +o(w)

)vk +o(w)(v) = ∑

j,k

∂ 2f (x)

∂x j∂xkw jvk +o(w)(v)

and so D2f (x)(w)(v) = ∑ j,k∂ 2f(x)∂x j∂xk

w jvk. Hence D2f is continuous if and only if each of

these coefficients x→ ∂ 2f(x)∂x j∂xk

is continuous. Obviously you can continue doing this and

conclude that Dkf is continuous if and only if all of the partial derivatives of order up to kare continuous. ■