In the case where X = ℝ^{n} there is a special notation which is often used to describe higher order mixed
partial derivatives. It is called multi-index notation.

Definition 17.7.1α = (α_{1},

⋅⋅⋅

,α_{n}) for α_{1}

⋅⋅⋅

α_{n}positive integers is called a multi-index.Forαamulti-index, |α|≡ α_{1} +

n ( n 2 )
= ∑ ( ∑ -∂-f (x)wj + o (w )) vk + o(w )(v)
k=1 j=1∂xj∂xk

∑ ∂2f (x)
= -------wjvk + o (w )(v)
j,k∂xj∂xk

and so

∑ 2
D2f (x)(w )(v ) = ∂-f (x)-wjvk
j,k ∂xj∂xk

Hence D^{2}f is continuous if and only if each of these coefficients

∂2f (x)
x → ∂xj∂xk

is continuous. Obviously you can continue doing this and conclude that D^{k}f is continuous if and only if all
of the partial derivatives of order up to k are continuous. ■

In practice, this is usually what people are thinking when they say that f is C^{k}. But as just argued, this
is the same as saying that the r linear form x → D^{r}f

(x )

is continuous into the appropriate space of linear
transformations for each r ≤ k.

Of course the above is based on the assumption that the first k derivatives exist and gives two
equivalent formulations which state that these derivatives are continuous. Can anything be said about the
existence of the derivatives based on the existence and continuity of the partial derivatives? This is in the
next section.