Page 40 - Vector Analysis

36 CHAPTER 2. Differentiation of Functions of Several Variables

4. If A Ď Rn is closed and bounded, and f : A Ñ Rm is continuous, then for each ε ą 0
we can choose δ depending only on ε such that

}f (x) ´ f (y)}Rm ă ε whenever }x ´ y}Rn ă δ and x, y P A .

The property (that δ can be chosen independent of the point x0) is called uniform
continuity.

Theorem 2.14. Let U Ď Rn be open, and f : U Ñ Rm be a vector-valued function. Then
the following assertions are equivalent:

1. f is continuous on U.

2. For each open set V Ď Rm, f ´1(V) Ď U is open, where f ´1(V) is the pre-image of V

under f defined by

f ´1(V) ” ␣x P U ˇ f (x) P V( .
ˇ

Proof. Before proceeding, we recall that B Ď f ´1(f (B)) for all B Ď U and f (f ´1(B)) Ď B
for all B Ď Rm.

“1 ñ 2” Let a P f ´1(V). Then f (a) P V. Since V is open in Rm, D εf(a) ą 0 such that

B(f (a), εf(a)) Ď V . By continuity of f (and Remark 2.13), there exists δa ą 0 such

that ( )( )

f B(a, δa) Ď B f (a), εf(a) .

Therefore, for each a P f ´1(V), D δa ą 0 such that

B(a, δa) Ď f ´1 ( ( (a, )) Ď f ´1(B(f (a), )) Ď f ´1(V) .
f B δa) εf (a)

Therefore, f ´1(V) is open.

“2 ñ 1” Let a P U and ε ą 0 be given. Define V = B(f (a), ε), then V is open. Since
a P f ´1(V) and f ´1(V) is open by assumption, there exists δ ą 0 such that B(a, δ) Ď
f ´1(V). Therefore,

() Ď f (f ´1(V)) Ď V = B(f (a), ε)
f B(a, δ)

which (with the help of Remark 2.13) implies that f is continuous at a. ˝