Real Analysis Proof (Limits of Functions)

In summary, the conversation discusses the existence of a sequence (xn) that converges to c, but the sequence (f(xn)) does not converge, given that the function f does not have a limit at c. The participants also consider the contrapositive of this claim and discuss the behavior of continuous functions on an open interval near an endpoint.
  • #1
jmjlt88
96
0
Let A be a subset of ℝ. Let c be a limit point of A. Consider the function f: A → ℝ

Claim: If the function f has not have a limit at c, then there exists a sequence (xn), where xn≠c for all n, such that lim xn=c, but the sequence (f(xn)) does not converge.

Since the function f does not have a limit at c, for all L in ℝ, there is some εL>0 such that for all δ>0 there is a point x in A satisfying 0<|x-c|<δ and yet |f(x)-L|≥εL.
I know how to create a sequence that converges to 0; I am having trouble making the sequence (f(xn)) not converge.

My initial idea was to pick points xn such that 0<|xn-c|<1/n and |f(xn)-n|≥εn for all n. However, this was not quite working out. Any bump in the right direction would be awesome!
 
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  • #2
Hint: the value f(c), if defined, is irrelevant.

I'm trying to hint without being obvious.
 
  • #3
Is f assumed to be continuous except at c?

In what ways can a continuous function on an open interval fail to have a limit at an endpoint?
 
  • #4
To pasmith, no, f is not assumed to be continuous.

To verty, indeed. A couple ways the sequence (f(xn)) could fail to converge is if we have some asymptotic behavior or something like the signum function. Unfortunately, the hint provided is not getting the wheels in motion. :(
 
  • #5
Think about the contrapositive of the claim, it may be easier to prove.
 
  • #6
The contrapositive of the claim would be:
"If for every sequence (xn) that converges to c, the sequence (f(xn)) converges, then f has a limit at c."

I tried the contrapositive (and also had some difficulty). However, I gave up rather quickly since I am so focus on constructing that sequence to prove the original claim. I would really like to move on (as I am only reviewing the major topics from introductory analysis). As such, another hint concerning the original claim would be fantastic!
 
  • #7
jmjlt88 said:
The contrapositive of the claim would be:
"If for every sequence (xn) that converges to c, the sequence (f(xn)) converges, then f has a limit at c."

I tried the contrapositive (and also had some difficulty).

How far did you get?

Did you show that the limit of the sequence [itex](f(x_n))[/itex] is independent of the sequence [itex](x_n)[/itex]?
 

FAQ: Real Analysis Proof (Limits of Functions)

1. What is a limit of a function in real analysis?

A limit of a function in real analysis is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It represents the value that a function approaches but does not necessarily reach at a specific input value.

2. How do you prove the limit of a function using the epsilon-delta definition?

To prove the limit of a function using the epsilon-delta definition, you must show that for any value of epsilon (a small distance from the limit) there exists a corresponding value of delta (a small distance from the input) such that when the input is within delta distance from the limit, the output of the function will be within epsilon distance from the limit.

3. What is the importance of continuity in real analysis?

Continuity is an important concept in real analysis because it ensures that the limit of a function exists and is equal to the function's value at that point. It also allows for the use of powerful theorems such as the Intermediate Value Theorem and the Mean Value Theorem.

4. How do you prove the continuity of a function using the epsilon-delta definition?

To prove the continuity of a function using the epsilon-delta definition, you must show that for any value of epsilon, there exists a corresponding value of delta such that when the input is within delta distance from a specific point, the output of the function will be within epsilon distance from the function's value at that point.

5. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function as the input approaches a specific value from one direction (either from the left or the right). A two-sided limit, on the other hand, considers the behavior of a function as the input approaches a specific value from both directions. In order for a two-sided limit to exist, the one-sided limits must be equal.

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