Page 47 - Vector Analysis

§2.4 Conditions for Differentiability 43

Moreover, if }x ´ a}Rn ă δ, then |kn| ď }k}Rn = }x ´ a}Rn ă δ ď δn; thus

ˇ Bf ˇ ε
B xn (a)knˇˇ ?
ˇf (a1 , ¨ ¨ ¨ , an´1, an + kn) ´ f (a1, ¨ ¨ ¨ , an) ´ ď |kn| .
ˇ n

As a consequence, if }x ´ a}Rn ă δ, by Cauchy’s inequality,

ˇ [ Bf Bf ]ˇ
ˇf (x) B x1 B xn
ˇ ´ f (a) ´ (a)(x1 ´ a1) + ¨ ¨ ¨ + (a)(xn ´ an) ˇ
ˇ

n
εÿ
ď ? |kj| ď ε}k}Rn = ε}x ´ a}Rn
n
j=1

which implies that f is differentiable at a. ˝

Remark 2.31. When two or more components of the Jacobian matrix [ B f ¨¨¨ Bf ] of a

B x1 B xn

scalar function f are discontinuous at a point x0 P U , in general f is not differentiable at x0.

For example, both components of the Jacobian matrix of the functions given in Example

2.27, 2.28, 2.44 are discontinuous at (0, 0), and these functions are not differentiable at
(0, 0).

Example 2.32. Let U = R2z␣(x, 0) P R2 ˇ x ě 0(, and f : U Ñ R be given by
ˇ

$ cos´1 x if y ą 0 ,

’ ax2 + y2 if y = 0 ,
’ if y ă 0 .
’ π
’
&
f (x, y) = arg(x + iy) =

’ 2π ´ cos´1 x
’ ax2 + y2
’
’
%

Then

$y if y ‰ 0 , $x if y ‰ 0 ,
B f (x, y) = & ´ x2 + y2
B f ’ x2 + y2
&

B x % 0 if y = 0 , and By (x, y) = 1

’ if y = 0 .
%
x

Since B f and B f are both continuous on U, f is differentiable on U.

Bx By

Definition 2.33. Let U Ď Rn be open, and f : U Ñ Rm be differentiable on U . f is
said to be continuously differentiable on U if the partial derivatives B fi exist and

B xj

are continuous on U for i = 1, ¨ ¨ ¨ , m and j = 1, ¨ ¨ ¨ , n. The collection of all continuously
differentiable functions from U to Rm is denoted by C 1(U ; Rm). The collection of all bounded
differentiable functions from U to Rm whose partial derivatives are continuous and bounded
is denoted by Cb1(U ; Rm).