Results 1 to 2 of 2

Math Help - help por favor, uniform continuity

  1. #1
    Newbie
    Joined
    Nov 2006
    Posts
    19

    help por favor, uniform continuity

    I have the function f [0, infinity) that is continuous on all R. I need to prove that f is uniformly continuous on [0, infinity) if either of the following are satisfied:
    a.) there exists an a>0 such that f is uniformly continuous on [a, inf).
    b.) the lim (as x approaches infinity) of f(x) exists as a real number.
    Thank you for any suggestions.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,793
    Thanks
    1688
    Awards
    1
    Being able to help you is dependent on what you know.
    Do you know that if a function f is continuous on a compact set then f is uniformly continuous there? Thus if a>0 then interval \left[ {0,a} \right] is compact (closed and bounded) so f is uniformly continuous there. Also we are given that f is uniformly continuous on [a,\infty ), but [0,\infty ) = \left[ {0,a} \right] \cup [a,\infty ).

    The part (b) is just a bit more difficult.
    Say that \lim _{x \to \infty } f(x) = r and \varepsilon  > 0.
    \left[ {\exists N \in Z^ +  } \right]\left( {x > N \Rightarrow \left| {f(x) - r} \right| < \frac{\varepsilon }{3}} \right).
    From above you can see that f is uniformly continuous on \left[ {0,N} \right].
    Now prove that f is uniformly continuous on [N,\infty ).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. uniform continuity of x^2
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 11th 2011, 04:17 PM
  2. Uniform continuity
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: April 9th 2010, 03:19 PM
  3. Uniform Continuity
    Posted in the Calculus Forum
    Replies: 12
    Last Post: January 24th 2010, 06:25 AM
  4. uniform differentiable => uniform continuity
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 30th 2009, 03:19 PM
  5. Uniform Continuity
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 15th 2009, 12:17 PM

Search Tags


/mathhelpforum @mathhelpforum