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Answers Linear Algebra

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Linear Algebra
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University of California Irvine
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4.4.8. A function f : ℝ → ℝ is said to be bounded at a point x
0
provided that there are positive
numbers ɛ and M so that |f (x)| < M for all x ( x
0
- ɛ, x
0
+ ɛ). Show that the set of points at which
a function is bounded is open. Let E be an arbitrary closed set. Is it possible to construct a
function f : ℝ → ℝ so that the set of points at which f is not bounded is precisely the set E?
a) Let


 be the set of all points where f is bounded.
Note 

 over each x,
This 


is bounded and exists, since each
is bounded.
Then if for every
we have, by our definition of a bounded function,

 and 
  
we have 
,
But this would mean that  
Which therefore means that x is an interior point of
. The conclusion follows that set
is
open.
Hence,
The set of points at which a function is bounded is open.
b) As the required function is unbounded, the pre-images of the function may or may not belong
to the arbitrary set E.
Hence, it is not possible to construct a function  so that the set of points at which f
is not bounded, is precisely the set E.
5.1.2 Prove the validity of the limit 

 
 .
Let .
Let
So, if
 
, then
 
 
 
  
 
    
 
Therefore, 

  
 .

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5.1.4. Prove the validity of the limit 

.
Let .
Let
 
 
.
Note that as
 
.
a) Suppose
 
, then
 
 
 
 
 
 
 
 
 
b) With that, if
 
, we have
 
 
   
 
 
 
 
 
 
 

This shows that for any there exists a such that
 
implies
 
. Thus, 

.
5.1.6. Recall that in the definition of 

 there is a requirement that x
0
be a point of
accumulation of the domain of f. Which values of x
0
would be excluded from consideration in
the limit 


 ?


 
 
If
, then
.
If
, then
is the imaginary part, that is with i (complex numbers).
If
, then
is positive and in the real part.

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4.4.8. A function f : ℝ → ℝ is said to be bounded at a point x0 provided that there are positive numbers ɛ and M so that |f (x)| < M for all x ∈ ( x0 - ɛ, x0 + ɛ). Show that the set of points at which a function is bounded is open. Let E be an arbitrary closed set. Is it possible to construct a function f : ℝ → ℝ so that the set of points at which f is not bounded is precisely the set E? a) Let 𝐵𝑓 = {𝑥: |𝑓(𝑥)| < 𝑀, 𝑀 > 0} be the set of all points where f is bounded. Note 𝑀 = sup({𝑀𝑥 : 𝑥 ∈ 𝑅, |𝑓(𝑥)| < 𝑀𝑥 }) over each x, This sup({𝑀𝑥 : 𝑥 ∈ 𝑅, |𝑓(𝑥) < 𝑀𝑥 }) is bounded and exists, since each 𝑀𝑥 is bounded. Then if for every 𝑥 ∈ 𝐵𝑓 we have, by our definition of a bounded function, ∀𝑥 ∈ 𝐵𝑓 ∃ 𝜖 > 0 and ∀𝑥 ∈ (𝑥 − 𝜖, 𝑥 + 𝜖) we have |𝑓(𝑥)| < ? ...
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