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Is it True that if f is Uniformly Continuous and Unbounded...Review my work please! 
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#1
Nov1610, 10:13 PM

P: 408

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>c}f(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>c}f(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>c}f(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? 


#2
Nov1710, 06:47 AM

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P: 18,346




#3
Nov1710, 10:06 AM

P: 408

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>c}f(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>c}f(x) = ±∞. Now suppose c is in [0,∞), then lim _{x>c}f(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. 


#4
Nov1710, 10:30 AM

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P: 18,346

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



#5
Nov1910, 11:59 AM

P: 408

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>c}f(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? 


#6
Nov1910, 12:39 PM

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P: 18,346

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
Nov1910, 03:32 PM

P: 352

You can be much cruder; try a piecewiselinear function of bounded slope.



#9
Nov1910, 07:40 PM

P: 408

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 ....



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