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Ok so I have figured out it is differentiable over the reals and its derivative is unbounded, now I'm not too sure whether this is uniformly continuous on the reals, I can use the lipschitz property and it seems like it isn't but I could be wrong. Any help will be very much appreciated.

Thanks

Raul
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1 Answers1

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Hint: this function has a finite limit at infinity, and is continuous on $[0,M]$ for any $M>0$.

detnvvp
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  • so it is uniformly continuous thanks, how did you work out the infinite limit if you dont mind me asking? Also, around when x is near 0 and even infinity, isnt the gradient unbounded? Is this like a counter example why non-uniform continuity >>>//>>>> Lipschitz? Thanks – Raul Oct 30 '13 at 02:29
  • ok I have shown its uniformly cts on the reals, what about in the interval [-1,1]? Thanks – Raul Oct 30 '13 at 02:57
  • If it is uniformly continuous on $\mathbb R$, then it is uniformly continuous on every subset of $\mathbb R$; you can see that from the definition. – detnvvp Oct 30 '13 at 03:06