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f(x ).
 
The matrix
f(x ) is called the Jacobian matrix of f at x . It is sometimes also
 
written as J(f; x ). Note that we have shown that if f is differentiable at x , then
 
each component function f has all ¬rst partial derivatives existing at x .
G 
T 11.1. L et D be an open set in RL and let f be a function de¬ned on D
with range in RK. If f is differentiable at a point x in D, then f is continuous at

x.

This follows immediately from (5) on letting h ; 0.
We will now show that the existence of partial derivatives at a point does
not imply differentiability. We shall consider real-valued functions f de¬ned on
R. For typographical convenience we shall represent points in the plane as
(x, y) instead of (x , x ).

Example 11.1. Let
xy
(x, y) " (0, 0),
,
x ; y
f (x, y) :
(x, y) : (0, 0).
0,

Since f (x, 0) : 0 for all x, *f /*x exists at (0, 0) and equals 0. Similarly
*f /*y" : 0. Along the line y : kx, the function f becomes
 
kx k
f (x, kx) : : x " 0,
,
x ; kx 1 ; k
26 TOPICS IN REAL ANALYSIS


f (x, kx) : k/(1 ; k). Thus lim f (x, y) does not exist.
and so lim
V V W   
Therefore f is not continuous at the origin and so by Theorem 11.1 cannot be
differentiable at the origin.

One might now ask, what if the function is continuous at a point and has
partial derivatives at the point? Must the function be differentiable at the
point? The answer is no, as the next example shows.

Example 11.2. Let
xy
(x, y) " (0, 0),
f (x, y) : (x ; y
(x, y) : 0.
0,

As in Example 11.1, *f /*x and *f /*y exist at the origin and are both equal
to zero. Note that

0 - ("x" 9 "y") : "x" 9 2"x" "y" ; "y"

implies that x ; y . 2"x" "y". Hence for (x, y) " 0

"xy" 1 (x ; y) 1
" f (x, y)" : - : (x ; y.
(x ; y 2 (x ; y 2

f (x, y) : 0, and so f is continuous at (0, 0).
Therefore lim
V W   
If f were differentiable at (0, 0), then for any h : (h , h )

f (0 ; h) : f (0) ; (
f(0), h2 ; (h),

where (h)/#h# ; 0 as h ; 0. Since
f (0) : ((*f /*x)(0), (*f /*y)(0)) : (0, 0) and
since f (0) : 0, we would have

f (h) : (h), h " 0.

From the de¬nition of f we get, writing (h , h ) in place of (x, y),

hh

(h) : .
((h ) ; (h )
 
We require that (h)/#h# ; 0 as h ; 0. But this is not true since

(h) hh

: ,
#h# (h ) ; (h )
 
and by Example 11.1, the right-hand side does not have a limit as h ; 0.
DIFFERENTIATION IN RL 27


A real-valued function f de¬ned on an open set D in RL is said to be of class
C I on D if all of the partial derivatives up to and including those of order k
exist and are continuous on D. A function f de¬ned on D with range in RK,
m 9 1, is said to be of class C I on D if each of its component functions
f , i : 1, . . . , m, is of class C I on D.
G
We saw that if
is a real-valued function de¬ned on an open interval D :
+t : :t : , in R, then
(t )h is a ˜˜good™™ approximation to
(t ;h)9
(t )
  
for small h at any point t in D at which
is differentiable. If
is of class C I

in D, then a better approximation can be obtained using Taylor™s theorem from
elementary calculus, one form of which states that for t in D and h such that

t ; h is in D

I\
G (t )
I (t )hI
 hG ;

(t ; h) 9
(t ) :  (6)
,
  i! k!
G
where
G denotes the ith derivative of
and t is a point lying between t and

t ; h. If k : 1, then the summation term in (6) is absent and we have a

restatement of the mean-value theorem. Since
I is continuous,


I (t ) :
I (t ) ; (h),

where (h) ; 0 as h ; 0. Substituting this relation into (6) gives

I
G (t )hG
 ; (h),

(t ; h) 9
(t ) :  (7)
  i!
G
where (h) : (h)hI/k !. Thus (h)/hI ; 0 as h ; 0. For small h, the polynomial
in (7) is thus a ˜˜good™™ approximation to
(t ; h) 9
(t ), in the sense that
 
the error committed in using the approximation tends to zero faster than hI.
We now generalize (7) to the case in which f is a real-valued function of
class C  or C  on an open set D in RL. We restrict our attention to functions
of class C  or C  because for functions of class C I with k 9 2 the statement
of the result is very cumbersome, and in this text we shall only need k : 1, 2.

T 11.2. L et D be an open set in RL, let x be a point in D, and let f be a

real-valued function de¬ned on D. L et h be a vector such that x ; h is in D.

(i) If f is of class C  on D, then

f (x ; h) 9 f (x ) : 1
f (x ), h2 ; (h), (8)
   
where (h)/#h# ; 0 as h ; 0.

(ii) If f is of class C  on D, then

f (x ; h) 9 f (x ) : 1
f (x ), h2 ; 1h, H(x )h2 ; (h), (9)

    
28 TOPICS IN REAL ANALYSIS


where (h)/#h# ; 0 as h ; 0 and


*f
H(x ) : (x ) . (10)
 *x *x 
HG

Proof. Let
(t) : f (x ; th). Then



(0) : f (x ) and
(1) : f (x ; h). (11)
 

It follows from the chain rule that
is a function of class C I on an open
interval containing the closed interval [0, 1] : +t : 0 - t - 1, in its interior
whenever f is of class C I . If f is of class C  , then using the chain rule, we get

L *f

(t) :  (x ; th)h : 1
f (x ; th), h2. (12)
*x  G 
G G

If f is of class C  , we get

L L *f

(t) :   (x ; th)h h : 1h, H(x ; th)h2, (13)
*x *x  GH 
G H H G

where H is the matrix de¬ned in (10).
Let f be of class C  . If we take k : 1, t : 0, and h : 1 in (6), recall that

the summation term in (6) is absent when k : 1, and use (11) and (12), we get

f (x ; h) 9 f (x ) :
(1) 9
(0) : 1
f (x ; t h), h2, (14)
  

where 0 : t : 1. Since f is of class C  , the function
f is continuous and thus


f (x ; t h) :
f (x ) ; (h),
  

where (h) ; 0 as h ; 0. Substituting this relation into (14) and using the

Cauchy”Schwarz inequality

"1 (h), h2" - # (h)# #h#
 

give (8), with (h) : 1 (h), h2.
 
If f is of class C  , we proceed in similar fashion. We take k : 2, t : 0, and

h : 1 in (6) and use (11), (12), and (13) to get

f (x ; h) 9 f (x ) : 1
f (x ), h2 ; 1h, H(x ; t h)h2, (15)

   
DIFFERENTIATION IN RL 29


where 0 : t : 1. Since f is of class C  , each entry of H is continuous. Hence

H(x ; t h) : H(x ) ; M(h),
 
where M(h) is a matrix with entries m (h) such that m (h) ; 0 as h ; 0.
GH GH
Substituting this relation into (15) and using the relations

"1h, M(h)h2" - #h# #M(h)h# - m(h)#h#,

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