5.8 Some Standard Notation
In the case where X = ℝn there is a special notation which is often used to describe higher
order mixed partial derivatives. It is called multi-index notation.
Definition 5.8.1 α = (α1,
) for α1
αn positive integers is called a
multi-index. For α a multi-index, |α|≡ α1
αn, and if x ∈ X,
and f a function, define
Then in this special case, the following is another description of what is meant by a Ck
Definition 5.8.2 Let U be an open subset of ℝn and let f : U → Y. Then for k
a nonnegative integer, a differentiable function f is Ck if for every
≤ k,Dαf exists
and is continuous.
Theorem 5.8.3 Let U be an open subset of ℝn and let f : U → Y. Then if
exists for r ≤ k, then Drf is continuous at x for r ≤ k if and only if Dαf is
continuous at x for each
Proof: First consider the case of a single derivative. Then as shown above, the matrix of
and to say that x → Df
is continuous is the same as saying that each of these partial
derivatives is continuous. Written out in more detail,
Now go to the second derivative.
Hence D2f is continuous if and only if each of these coefficients
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 k are continuous.
In practice, this is usually what people are thinking when they say that f is Ck. But as
just argued, this is the same as saying that the r linear form x → Drf
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.