1
3.1 Limits
Recall:
Definition:
A vector-valued function f:
ℝ
n
→
ℝ
m
is a rule or process that assigns to each input
x
in
ℝ
n
corresponding output
y
in
ℝ
m
if m > 1.
A real function f of n variables is a rule that assigns to each input
x
or n-tuple of real numbers
(x
1
, x
2
, …, x
n
) in a domain D
⊂
ℝ
n
a unique real number, denoted f (x
1
, x
2
, …, x
n
).
So a real function f of n variables is the same as a real function of a vector
x
= [x
1
, x
2
, …, x
n
], or
a real value function of a point (x
1
, x
2
, …, x
n
).
Let
x
0
be a point in
ℝ
n
.
Open disk
or
open ball
D
r
(
x
0
) = {
x
∈
ℝ
n
| ||
x
–
x
0
|| < r}
⊂
ℝ
n
.
Definition: Let U
⊂
ℝ
n
. For each point
x
0
in U, there exists some r > 0 such that D
r
(
x
0
)
⊂
U.
Then U is called as an
open set
of
ℝ
n
.
Theorem: D
r
(
x
0
) is an open set.