Is it True that if f is Uniformly Continuous and Unbounded Review my work please

In summary, if f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞.
  • #1
mmmboh
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[PLAIN]http://img820.imageshack.us/img820/7729/3iiin.jpg [Broken]

So I gave it a go, and I just want to make sure my argument is convincing:

If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞(notice the closed brackets, I wanted to leave the option that c can be infinity, is this how I should write it? if not, how?), because if this wasn't true, then lim x->cf(x)=A (A is finite) for all c in [0,∞], but then f would be bounded (is this convincing?). Now suppose c is in [0,∞) (open bracket, so c isn't infinity), then lim x->cf(x) = ±∞, but since c is finite and f is continuous, then by the definition of continuity f is defined at c, but then f(c)= ±∞, but ±∞ isn't even in the reals so then f wouldn't be undefined at c, so it wouldn't be continuous. So c=∞, and lim x->∞f(x) = ±∞.

Thoughts?
 
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  • #2


mmmboh said:
because if this wasn't true, then lim x->cf(x)=A (A is finite) for all c in [0,∞], but then f would be bounded (is this convincing?)

I disagree with this. If the limit is not infinity, then the possibility also exists that the limit does not exist. For example [tex]f(x)=x\sin(x)[/tex] (note that this function is not uniform continuous, so it doesn't contradict our statement.
 
  • #3


Ok well, the limit may be not exist at infinity, but it will still be bounded, no? (if it is uniformly continuous). How do you think I should approach this?

Note: If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞. This still remains true right? I mean this is the definition of unbounded, so maybe I can just erase the part you disagreed with?

Basically if I wrote this: If f is uniformly continuous and unbounded on [0,∞), then for some c in [0,∞], lim x->cf(x) = ±∞. Now suppose c is in [0,∞), then lim x->cf(x) = ±∞, but since c is finite and f is continuous, then by the definition of continuity f is defined at c, but then f(c)= ±∞, but ±∞ isn't even in the reals so then f wouldn't be undefined at c, so it wouldn't be continuous. So c=∞, and lim x->∞f(x) = ±∞.

Edit: Hm come to think of it, what I wrote might imply that lim x->∞xsinx=∞ (ignoring the uniformly continuous part), so I need to justify it for uniform continuity somehow, assuming it is true.
 
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  • #4


mmmboh;2990330 Note: If f is uniformly continuous and unbounded on [0 said:
, lim x->cf(x) = ±∞. This still remains true right?

Hmm, I think that you should prove this. I don't think it's evident (I don't even know if it's true)...
 
  • #5


Well that's almost what the question is asking me to prove. Do you have an idea on how to do it for me? I know that lim x->cf(x) = ±∞ could not happen anywhere but at infinity, but I'm not sure how to show it must happen at infinity in the case of uniform continuity (assuming it is true, I believe it is).

I guess I can show that if f is unbounded and continuous on [0,∞) and lim x->∞f(x)≠±∞, then it is not uniformly continuous, but still, any ideas?
 
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  • #6


If you don't find anything fast, then maybe you can start searching after counterexamples...
A counterexample will look like f(x)=xsin(x), but this function goes to fast up and down. Maybe you can find a function which will go up and down a bit slower...
 
  • #7


You can be much cruder; try a piecewise-linear function of bounded slope.
 
  • #8


Yes, that's a very good counterexample!
 
  • #9


Thanks. You think it's ok to just leave it defined as I did? because I can't think of a nicer way to do it, you know without having to write ...
 
  • #10


Well, I think the idea of the counterexample is clear. You could probably make the definition more formal, but this would be rather ugly and imo doesn't help in grasping the concept. So I'd say to leave it like it is...
 

1. Is it true that if f is uniformly continuous, it must also be unbounded?

No, this statement is not necessarily true. Uniform continuity is a property of a function that describes its behavior near a point, while being unbounded means that the function has no limit as the input approaches infinity. A function can be uniformly continuous without being unbounded, and vice versa.

2. Can you explain the concept of uniform continuity in simpler terms?

Uniform continuity means that for any given small change in the input, there exists a small enough change in the output that ensures the function stays close to its original value. In other words, the function doesn't have any sudden or large jumps in its values as the input changes.

3. How does uniform continuity differ from regular continuity?

Uniform continuity is a stronger condition than regular continuity. While regular continuity guarantees that the function is continuous at every point, uniform continuity also requires that the function doesn't have any sudden jumps in its values as the input changes.

4. What are some examples of functions that are uniformly continuous and unbounded?

One example is the function f(x) = x^2, which is uniformly continuous but unbounded as the input approaches infinity. Another example is the function f(x) = sin(x), which is uniformly continuous but unbounded as the input oscillates between -1 and 1.

5. Are there any real-life applications of uniform continuity and unboundedness?

Yes, these concepts are important in mathematics and engineering, particularly in analysis and optimization problems. For example, when studying the behavior of a system, it is useful to know if a function describing the system is uniformly continuous and unbounded, as it can provide insights into its behavior and help in finding optimal solutions.

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