5.13 C1 Functions
However, there are theorems which can be used to get differentiability of a function based
on existence of the partial derivatives.
Definition 5.13.1 When all the partial derivatives exist and are continuous the
function is called a C1 function.
Because of Proposition 5.11.4 on Page 331 and Theorem 5.12.5 which identifies the
entries of Jf with the partial derivatives, the following definition is equivalent to the
Definition 5.13.2 Let U ⊆ Fn be an open set. Then f : U → Fm is C1
if f is
differentiable and the mapping
is continuous as a function from U to ℒ
The following is an important abstract generalization of the familiar concept of partial
Definition 5.13.3 Let g : U ⊆ Fn × Fm → Fq, where U is an open set in Fn × Fm.
Denote an element of Fn × Fm by
where x ∈ Fn and y ∈ Fm. Then the map
x → g
is a function from the open set in Fn,
to Fq. When this map is differentiable, its derivative is denoted by
A similar definition holds for the symbol Dyg or D2g. The special case seen in beginning
calculus courses is where g : U → Fq and
The following theorem will be very useful in much of what follows. It is a version of
the mean value theorem. You might call it the mean value inequality.
Theorem 5.13.4 Suppose U is an open subset of Fn and f : U → Fm has the property
exists for all x in U and that, x
∈ U for all t ∈
. (The line
segment joining the two points lies in U.) Suppose also that for all points on this line
Then 0 ∈ S and by continuity of f, it follows that if t ≡ supS, then t ∈ S and if
t < 1,
If t < 1, then there exists a sequence of positive numbers,
converging to 0
which implies that
By 5.13.18, this inequality implies
which yields upon dividing by hk and taking the limit as hk → 0,
Now by the definition of the norm of a linear operator,
a contradiction. Therefore, t
= 1 and so
Since ε > 0 is arbitrary, this proves the theorem.
The next theorem proves that if the partial derivatives exist and are continuous, then
the function is differentiable.
Theorem 5.13.5 Let g : U ⊆ Fn × Fm → Fq. Then g is C1
if and only if D1g and
D2g both exist and are continuous on U. In this case,
Proof: Suppose first that g ∈ C1
. Then if
A similar argument applies for D2g and this proves the continuity of the function,
2. The formula follows from
Now suppose D1g
exist and are continuous.
. Then the expression in
is of the
and so, by continuity of
is small enough. By Theorem
on Page 347
, there exists δ >
such that if
, the norm of the last term in 5.13.19
Therefore, this term is o
. It follows from
Showing that Dg
exists and is given by
The continuity of
follows from the continuity of
This proves the theorem.
Not surprisingly, it can be generalized to many more factors.
Definition 5.13.6 Let g : U ⊆∏
i=1nFri → Fq, where U is an open set. Then the map
xi → g
is a function from the open set in Fri,
to Fq. When this map is differentiable, its derivative is denoted by Dig
. To aid in the
notation, for v ∈ Fri, let θiv ∈∏i=1nFri be the vector
where the v is in
the ith slot and for v ∈∏
i=1nFri, let vi denote the entry in the ith slot of v.
Thus, by saying xi → g
is differentiable is meant that for v ∈ Fri sufficiently
Here is a generalization of Theorem 5.13.5.
Theorem 5.13.7 Let g,U,∏
i=1nFri, be given as in Definition 5.13.6. Then g is
if and only if Dig exists and is continuous on U for each i. In this
where v =
Proof: Suppose then that Dig exists and is continuous for each i. Note that
and define ∑
j=10θjvj ≡ 0
Consider the terms in this sum.
and the expression in 5.13.24 is of the form h
where for small
w ∈ Frk,
and by continuity,
is small enough. Therefore, by
is small enough,
which shows that since
is arbitrary, the expression in 5.13.24
. Now in
Therefore, referring to 5.13.22,
which shows Dg exists and equals the formula given in 5.13.21.
Next suppose g is C1. I need to verify that Dkg
exists and is continuous. Let
v ∈ Frk
sufficiently small. Then
exists and equals
Since x → Dg
is continuous and
: Frk →∏
is also continuous, this proves
The way this is usually used is in the following corollary, a case of Theorem 5.13.7
obtained by letting Frj = F in the above theorem.
Corollary 5.13.8 Let U be an open subset of Fn and let f :U → Fm be C1 in the sense
that all the partial derivatives of f exist and are continuous. Then f is differentiable
class=”left” align=”middle”(Fn, Fm)5.14. CK FUNCTIONS