Page 38 - Vector Analysis

34 CHAPTER 2. Differentiation of Functions of Several Variables

1. A is open if and only if A = A˚;

2. A is closed if and only if A = A;

3. A is closed if and only if B A Ď A.

Definition 2.5 (Level Sets, and Graphs). Let A Ñ Rn be a set, and f : A Ñ R be a

real-valued function. The collection of points in A where f has a constant value is called a
( )
level set of f . The collection of all points x, f (x) is called the graph of f .

Remark 2.6. A level surface is conventionally called a level curve when n = 2.

2.2 Limits and Continuity

Definition 2.7. Let A Ď Rn be a set, and f : A Ñ Rm be a vector-valued function. For a
given x0 P A1, we say that b P Rm is the limit of f at x0, written

lim f (x) = b or f (x) Ñ b as x Ñ x0 ,

xÑx0

if for each ε ą 0, there exists δ = δ(x0, ε) ą 0 such that

}f (x) ´ b}Rm ă ε whenever 0 ă }x ´ x0}Rn ă δ and x P A .

By the definition above, it is easy to see the following

Proposition 2.8. Let A Ď Rn be a set, and f, g : A Ñ Rm be a vector-valued functions.

Suppose that x0 P A1, f (x) = g(x) for all x P Aztx0u, and lim f (x) exists. Then lim g(x)
xÑx0 xÑx0
exists and

lim g(x) = lim f (x) .
xÑx0 xÑx0

The following proposition is standard, and we omit the proof.

Proposition 2.9. Let A Ď Rn be a set, and f, g : A Ñ Rm be vector-valued functions,

h : A Ñ R be a real-valued function. Suppose that x0 P A1, and lim f (x) = a, lim g(x) = b,
xÑx0 xÑx0

lim h(x) = c. Then
xÑx0

lim (f + g)(x) = a + b , lim (f ´ g)(x) = a ´ b ,

xÑx0 xÑx0

lim (hf )(x) = ca , lim (f ¨ g)(x) = a ¨ b ,
xÑx0
xÑx0

lim ( f ) = a if c ‰ 0 .
xÑx0 h c