Page 62 - Vector Analysis

58 CHAPTER 2. Differentiation of Functions of Several Variables

As a consequence, if k is such that b + k P V,

››f ´1 (b + k ) ´ f ´1 (b) ´ ( )(a))´1 k › ››a + h ´ a ´ ( )(a))´1k››Rn
(Df ›Rn (Df
=
}k}Rn }k}Rn

ď ››((Df )(a))´1››B(Rn,Rn) ››k ´ (Df )(a)(h)››Rn
}k}Rn

ď ››((Df )(a))´1››B(Rn,Rn) ››f (a + h) ´ f (a) ´ (Df )(a)(h)››Rn }h}Rn
ď ››((Df )(a))´1››B(Rn,Rn) }h}Rn }k}Rn

››f (a + h) ´ f (a) ´ (Df )(a)(h)››Rn .

λ }h}Rn

Using (2.10), h Ñ 0 as k Ñ 0; thus passing k Ñ 0 on the left-hand side of the inequality

above, by the differentiability of f we conclude that

››f ´1 (b + k ) ´ f ´1 (b) ´ ( )(a))´1k››Rn
(Df
lim = 0.
kÑ0 }k}Rn

This proves 3. ˝

Remark 2.62. Since f ´1 : V Ñ U is continuous, for any open subset W of U f (W) =
(f ´1)´1(W) is open relative to V, or f (W) = O X V for some open set O Ď Rn. In other
words, if U is an open neighborhood of x0 given by the inverse function theorem, then
f (W) is also open for all open subsets W of U. We call this property as f is a local open
mapping at x0.

Remark 2.63. Since (Df )(x0) P B(Rn, Rn), the condition that (Df )(x0) is invertible can
be replaced by that the determinant of the Jacobian matrix of f at x0 is not zero; that is,

([ ])
det (Df )(x0) ‰ 0 .

The determinant of the Jacobian matrix of f at x0 is called the Jacobian of f at x0. The

Jacobian of f at x sometimes is denoted by Jf (x) or B (f1, ¨ ¨ ¨ , fn) .
B (x1, ¨ ¨ ¨
, xn)

Example 2.64. Let f : R Ñ R be given by

f (x) = # x + 2x2 sin 1 if x ‰ 0 ,
x

0 if x = 0 .

Let 0 P (a, b) for some (small) open interval (a, b). Since f 1(x) = 1 ´ 2 cos 1 + 4x sin 1 for
x x
x ‰ 0, f has infinitely many critical points in (a, b), and (for whatever reasons) these critical