17.5 Existence Of The Derivative, C1 Functions
There is a way to get the differentiability of a function from the existence and continuity of the Gateaux
derivatives. This is very convenient because these Gateaux derivatives are taken with respect to a one
dimensional variable. The following theorem is the main result.
Theorem 17.5.1 Let X be a normed vector space having basis
and let Y be another normed
vector space having basis
. Let U be an open set in X and let f
: U → Y have the property
that the Gateaux derivatives,
exist and are continuous functions of x. Then Df
Furthermore, x → Df
is continuous; that is
Proof: Let v =∑
Then letting ∑
k=10 ≡ 0, f
is given by
Consider the kth term in 17.6. Let
for t ∈
and this equals
Now without loss of generality, it can be assumed that the norm on X is given by
because by Theorem 11.5.4 all norms on X are equivalent. Therefore, from 17.7 and the assumption that
the Gateaux derivatives are continuous,
is sufficiently small. Since
is arbitrary, it follows from Lemma 17.4.1
the expression in 17.6
because this expression equals a finite sum of terms of the form
is small enough. Thus
Consider the kth term in the second sum.
where the expression in the parentheses converges to 0 as ak → 0. Thus whenever
which shows the second sum is also o
where v = ∑
kakvk, it follows Df
and is given by the above formula.
It remains to verify x → Df
which proves the continuity of Df because of the assumption the Gateaux derivatives are continuous.
This motivates the following definition of what it means for a function to be C1.
Definition 17.5.2 Let U be an open subset of a normed finite dimensional vector space, X and let
f : U → Y another finite dimensional normed vector space. Then f is said to be C1 if there exists a basis for
such that the Gateaux derivatives,
exist on U and are continuous.
Note that as a special case where X = ℝn, you could let the vk = ek and the condition would reduce to
nothing more than a statement that the partial derivatives
are all continuous.
Here is another definition of what it means for a function to be C1.
Definition 17.5.3 Let U be an open subset of a normed vector space, X and let f : U → Y another
normed vector space. Then f is said to be C1 if f is differentiable and x → Df
is continuous as
a map from U to ℒ
Now the following major theorem states these two definitions are equivalent. This is obviously so in the
special case where X = ℝn and the special basis is the usual one because, as observed earlier, the matrix of
is just the one which has for its columns the partial derivatives which are given to be
Theorem 17.5.4 Let U be an open subset of a normed finite dimensional vector space, X and
let f : U → Y another finite dimensional normed vector space. Then the two definitions above are
Proof: It was shown in Theorem 17.5.1, the one about the continuity of the Gateaux derivatives
yielding differentiability, that Definition 17.5.2 implies 17.5.3. Suppose then that Definition 17.5.3 holds.
Then if v is any vector,
exists and equals
By continuity of x → Df
this establishes continuity of
x → Dvf
and proves the theorem.
Note that the proof of the theorem also implies the following corollary.
Corollary 17.5.5 Let U be an open subset of a normed finite dimensional vector space, X and let
f : U → Y another finite dimensional normed vector space. Then if there is a basis of X,
such that the Gateaux derivatives, Dvkf
exist and are continuous. Then all Gateaux derivatives,
exist and are continuous for all v ∈ X.
From now on, whichever definition is more convenient will be used.