11.1 Review Of Limits
Recall the concept of limit of a function of many variables. When f : D
one can only consider in a
meaningful way limits at limit points of the set D
Definition 11.1.1 Let A denote a nonempty subset of ℝp. A point x is said to be a limit point
of the set A if for every r > 0,B
contains infinitely many points of A.
Example 11.1.2 Let S denote the set
. Which points are limit
This set does not have any because any two of these points are at least as far apart as 1. Therefore, if x
is any point of ℝ3,B
contains at most one point.
Example 11.1.3 Let U be an open set in ℝ3. Which points of U are limit points of U?
They all are. From the definition of U being open, if x ∈ U, There exists B
for some r >
Now consider the line segment x
where t ∈
. This describes infinitely many points and they
are all in
tr < r.
Therefore, every point of U
is a limit point of
The case where U is open will be the one of most interest, but many other sets have limit
Definition 11.1.4 Let f : D
⊆ ℝp → ℝq where q,p ≥
1be a function and let x be a limit point of
if and only if the following condition holds. For all ε > 0 there exists δ > 0 such that if
The condition that x must be a limit point of D
if you are to take a limit at
is what makes the
limit well defined.
Proposition 11.1.5 Let f : D
⊆ ℝp → ℝq where q,p ≥
1be a function and let x be a limit point
. Then if
exists, it must be unique.
Proof: Suppose limy→xf
. Then for ε >
0 given, let δi >
in the definition of the limit and let δ
is a limit point, there exists
y ∈ B
Since ε > 0 is arbitrary, this shows L1 = L2. ■
The following theorem summarized many important interactions involving continuity. Most of this
theorem has been proved in Theorem 8.5.5 on Page 419.
Theorem 11.1.6 Suppose x is a limit point of D
where K and L are vectors in ℝp for p ≥ 1. Then if a, b ∈ ℝ,
Also, if h is a continuous function defined near L, then
For a vector valued function
T if and only if
for each k = 1,
In the case where f and g have values in ℝ3
Also recall Theorem 8.5.6 on Page 428.
Theorem 11.1.7 For f : D
→ ℝq and x ∈ D
such that x is a limit point of D
, it follows
f is continuous at x if and only if