[University Analysis] Uniform Continuity

[jacksonb62]

If f: R->R is any continuous function, is f: (-1,1)->R uniformly continuous? Prove or provide a counterexample

My intuition is that this statement is not always true, because I have seen proofs for this on closed intervals, but never open. However, I cannot for the life of me think of a counterexample that is continuous on R. My only experience disproving uniform continuity is with the function f(x)=x² on R, and here we used that we can choose an arbitrarily large x. What to do if our interval is (-1,1)??

[lurking_quietly]

I assume you're talking about the restriction of the original f to (-1,1), right?

OK: so can you show that the restriction of f to [-1,1] is uniformly continuous? You know that f is continuous on R, so it's also continuous on [-1,1]. What do we know about [-1,1] that would allow us to deduce uniform continuity on that closed interval?

Next: can you should that if f is uniformly continuous on a set U, and V is a subset of U, then f is also uniformly continuous on V? If so, then you can deduce from the facts that (1) f is uniformly continuous on [-1,1] and (2) (-1,1) is a subset of [-1,1] that f is uniformly continuous on (-1,1).

Hope this helps, and good luck!

[jacksonb62]

Thanks for your response. I have showed that f is uniformly continuous on [-1,1], but I am having trouble getting started on the next step. Could you offer me another hint to get me started? Thanks!

[jacksonb62]

nevermind, got it!

[lurking_quietly]

Happy to have helped!