Need example of a continuous function map cauchy sequence to non-cauchy sequence

In summary, a continuous function f:(X, d) -> Y(Y, p) does NOT always map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces, as demonstrated by the counterexample of f:R-{0} -> R-{0} defined by f(x)= 1/x. This function is continuous on its domain, but does not map a Cauchy sequence in R-{0} to a Cauchy sequence in R-{0} of its images. This example shows that not all continuous functions can preserve Cauchy sequences between complex planes in metric spaces.
  • #1
xsw001
37
0

Homework Statement



I need a example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to a Cauchy sequence of its images [f(xn) in Y] in the complex plane between metric spaces.

Homework Equations



If a function f is continuous in metric space (X, d), then it continuous at every point in X.

Definition of a sequence converges to a point in metric space:
If {xn} -> x, then for all e>0, there exists an N such that d(x, xn)<e, for all n<= N

Definition of Cauchy sequence in metric space:
- {xn} is a Cauchy sequence if for all e>0, there exists an N such that d(xn, xm)<e, for all n, m <= N
- Every convergent sequence in metric space is a Cauchy sequence.

The Attempt at a Solution



I have counterexample in mind in R but not complex plane in metric space though.

Let f:R-{0} -> R-{0} defined by f(x)= 1/x, so f(x) is continuous on all domain.
Let {xn} = 1/n, then {xn} converges to 0 and therefore is a Cauchy sequence in R–{0}, but its image f(xn) = 1/xn = 1/(1/n) = n diverges, therefore is NOT a Cauchy sequence.

But I need an example of a continuous function f:(X, d) -> Y(Y, p) does NOT map a Cauchy sequence [xn in X] to the Cauchy sequence of its images [f(xn) in Y] between complex planes in the metric spaces?

Any suggestions?
 
Physics news on Phys.org
  • #2
1/x?

EDIT: nm i didnt read the whole post
 

1. What is a continuous function?

A continuous function is a function in which small changes in the input result in small changes in the output. In other words, as the input values get closer, the output values also get closer. This property is known as continuity.

2. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers in which the terms get closer and closer together as the sequence progresses. In other words, for any small distance, there is a point in the sequence where all subsequent terms are within that distance of each other.

3. How can a continuous function map a Cauchy sequence to a non-Cauchy sequence?

A continuous function can map a Cauchy sequence to a non-Cauchy sequence by introducing a "bump" or "jump" in the graph of the function. This can cause the output values to suddenly increase or decrease, resulting in a non-Cauchy sequence.

4. Why is it important to understand continuous functions and Cauchy sequences?

Understanding continuous functions and Cauchy sequences is important in the study of calculus and analysis. These concepts allow us to understand the behavior of functions and sequences, and their properties play a crucial role in many mathematical proofs and applications.

5. Can you provide an example of a continuous function that maps a Cauchy sequence to a non-Cauchy sequence?

Yes, one example is the function f(x) = sin(1/x). As x approaches 0, the function oscillates infinitely between -1 and 1, causing the output values to suddenly increase or decrease. This results in a non-Cauchy sequence, even though the input sequence may be Cauchy.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
701
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
27
Views
692
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
Replies
1
Views
564
  • Calculus and Beyond Homework Help
Replies
13
Views
943
  • Calculus and Beyond Homework Help
Replies
1
Views
214
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
28
Views
2K
Back
Top