Problem about uniformly continuous

  • Thread starter herbyoung
  • Start date
  • Tags
    Continuous
In summary, the conversation discusses a proof by contradiction for the statement that there exist positive constants A and B such that |f(x)|<=Ax+B for all x in the interval I=[0,infinity), assuming that f is uniformly continuous. The conversation also hints at using the definition of uniform continuity and the reverse triangle inequality to derive a contradiction.
  • #1
herbyoung
3
0

Homework Statement



Let I be the interval I=[0,infinity). Let f: I to R be uniformly continuous. Show there exist positive constants A and B such that |f(x)|<=Ax+B for all x that belongs to I.


2. The attempt at a solution
Proof by contradiction.
 
Physics news on Phys.org
  • #2
Proof by contradiction:
Let I be the interval I = [0, infinity). Let f: I to R be uniformly continuous.

Suppose that for all positive A and B, there is an x = x(A, B) in I such that |f(x)| > Ax + B.

All you have to derive a contradiction with is the uniform continuity of f. Can you write this out in terms of f(x) (i.e. the definition)?
 
  • #3
Can you complete the answer please ??
I am interesting to know how we can prove this .
 
  • #4
I didn't solve it myself at the time, but some quick scribbles show that if you use the definition and the reverse triangle inequality it should work out.
How far did you get?
 
  • #5
mariama1 said:
Can you complete the answer please ??

We don't provide complete answers here, only hints. If you're trying to do this problem yourself, please start a new thread instead of hijacking an old one.
 

1. What is the definition of uniform continuity?

Uniform continuity is a property of a function where the change in the output value is always proportional to the change in the input value, regardless of the size of the interval. In other words, the function remains continuous over the entire domain.

2. How is uniform continuity different from regular continuity?

Uniform continuity differs from regular continuity in that it guarantees that the function will remain continuous over the entire domain, rather than just at a specific point. This means that the function will not have any sudden jumps or breaks in its graph.

3. What is the importance of uniform continuity?

Uniform continuity is important because it ensures the stability of a function over its entire domain. This allows for the prediction of the function's behavior at any given point, making it useful in various mathematical and scientific applications.

4. What are the necessary conditions for a function to be uniformly continuous?

In order for a function to be uniformly continuous, it must be continuous over its entire domain and have a bounded derivative. This means that the rate of change of the function cannot exceed a certain value, ensuring that the function does not have any sudden changes in its behavior.

5. Can a function be uniformly continuous but not differentiable?

Yes, a function can be uniformly continuous but not differentiable. This means that the function is continuous over its entire domain, but the derivative does not exist at certain points. This can occur when the function has sharp corners or cusps, which violate the condition of having a bounded derivative.

Similar threads

  • Calculus and Beyond Homework Help
Replies
26
Views
804
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
129
  • Calculus and Beyond Homework Help
Replies
26
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
223
  • Calculus and Beyond Homework Help
Replies
1
Views
537
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Math POTW for University Students
Replies
15
Views
872
Back
Top