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 VW

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

VW

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#,