1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Uniform continuity and bounded

  1. Nov 8, 2005 #1
    Prove that if f is uniformly continuous on a bounded set S then f is bounded on S.

    Our book says uniform continuity on an interval implies regular continuity on the interval, and in the previous chapter we proved that if a function is continuous on some closed interval then it is bounded. Is that how I should prove this, or am I missing something? It seems to easy this way and that usually means I am overlooking something.


    tia
     
  2. jcsd
  3. Nov 8, 2005 #2
    You're overlooking the fact that not all bounded sets are intervals.

    A possible way to prove the result, is to use that a uniformly continuous function maps Cauchy sequences to Cauchy sequences. If you assume that f is unbounded on S, this will lead to a contradiction.
     
  4. Nov 8, 2005 #3
    I was just getting ready to say just because S is bounded doesnt mean S is closed and I need S to be closed to go the easy way. I supposed f is uniformly continuous and unbounded then I said since S is bounded then there is a convergent sequence {Xn} in S by bolzano and since {Xn} converges then {Xn} is cauchy. Since f is uniformly continuous on S then f(Xn) is cauchy so f is bounded, contradiction
     
  5. Nov 8, 2005 #4
    That's not correct, but on the right path. You don't actually have a contradiction (unbounded sets can (and always do!) contain Cauchy sequences). You need to place more requirements on x_n.
     
  6. Nov 8, 2005 #5
    what kind of requirements do I need? Do I need to say something about Xn converging to an element in S? Do I need to use subsequences?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Uniform continuity and bounded
Loading...