Page 48 - Vector Analysis

44 CHAPTER 2. Differentiation of Functions of Several Variables

Example 2.34. If f : R Ñ R is differentiable at x0, must f 1 be continuous at x0? In other

words, is it always true that lim f 1(x) = f 1(x0)?
xÑx0

Answer: No! For example, take

$ x2 sin 1 if x ‰ 0,
&

f (x) = x

% 0 if x = 0.

1˝ Show f (x) is differentiable at x = 0:

f 1(0) = lim f (0 + h) ´ f (0) = lim h2 sin 1 = lim h sin 1 = 0.
h

hÑ0 h hÑ0 h hÑ0 h

2˝ We compute the derivative of f and find that

$ 2x sin 1 ´ cos 1 if x ‰ 0,
& x x

f 1(x) =

%0 if x = 0.

However, lim f 1(x) does not exist.
xÑ0

Definition 2.35. Let U Ď Rn be open, and f : U Ñ R be a function. If the partial

derivative B f exists in U and has partial derivatives (at every point in U) with respect to
B xj B ( Bf )
xi, then partial derivatives is denoted by B2f .
the second-order B xi B xj
B xiB xj

In general, if the k-th order partial derivatives B kf exists in U and has

B xik B xik´1 ¨ ¨ ¨ B xi1

partial derivatives (at every point in U) with respect to xik+1 , then the (k + 1)-th order
partial derivatives ( ) ; that is,
B Bkf B k+1f
is denoted by
B xik+1 B xik B xik´1 ¨ ¨ ¨ B xi1 B xik+1 B xik ¨ ¨ ¨ B xi1

B k+1f = B( Bkf )
.
B xik+1 B xik ¨ ¨ ¨ B xi1 B xik+1 B xik B xik´1 ¨ ¨ ¨ B xi1

Theorem 2.36. Let U Ď Rn be open, a P U, and f : U Ñ R be a real-valued function.

Suppose that for some 1 ď i, j ď n, Bf , Bf , B 2f and B 2f exist in a neighborhood
B xjB xi B xiB xj
B xi B xj

of a and are continuous at a. Then

B 2f (a) = B 2f (a) .
B xiB xj B xjB xi