Results 1 to 4 of 4

Math Help - bounded continuous function

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    43

    bounded continuous function

    must a bounded continuous function on R be uniformly continuous?
    I know that if a function is continuous on a closed and bdd set then its uniformly continuous, but this says nothing about the set, just the function. Would this be T or False? How would I go about proviing it?

    Also, if f and g are uniformly continuous maps of R to R, must the product f*g be uniformly continuous? What if f and g are bounded.
    I think this one is true...since you're just multiplying two continuous functions together...but I don't know how to prove this either

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by cp05 View Post
    must a bounded continuous function on R be uniformly continuous?
    I know that if a function is continuous on a closed and bdd set then its uniformly continuous, but this says nothing about the set, just the function. Would this be T or False? How would I go about proviing it?
    Well, for (0,1) what about f(x)=\sin\left(\frac{1}{x}\right)? That is bounded and continuous but not uniformly continuous. Can you see how to generalize?

    Also, if f and g are uniformly continuous maps of R to R, must the product f*g be uniformly continuous? What if f and g are bounded.
    I think this one is true...since you're just multiplying two continuous functions together...but I don't know how to prove this either

    Thanks.
    It isn't true, take f(x)=g(x)=x
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    43
    why is not uniformly continuous? aren't all the y-values relatively close together? I don't really understand how to prove or disprove uniform continuity other than saying when the x values are really close together so are the y values...

    for the second part, what about when f and g are bounded? i know when the derivative is bounded they are uniform continuous because it will be a lipschitz function, but I haven't read anything about f and g being bounded
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by cp05 View Post
    for the second part, what about when f and g are bounded? i know when the derivative is bounded they are uniform continuous because it will be a lipschitz function, but I haven't read anything about f and g being bounded
    If f and g are both uniformly continuous on R, and are both bounded, then fg will be uniformly continuous, because

    \begin{aligned}|f(x)g(x)-f(y)g(y)| &= |f(x)\bigl(g(x)-g(y)\bigr) + \bigl(f(x)-f(y)\bigr)g(y)| \\ &\leqslant |f(x)||g(x)-g(y)| + |f(x)-f(y)||g(y)| \end{aligned}<br />

    and you can make the right-hand side of that inequality as small as you want, for x and y sufficiently close together, using the given properties of f and g.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] right limit of a continuous bounded function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: June 8th 2011, 07:09 AM
  2. Function Bounded and Continuous on (0,1) but not Integrable!
    Posted in the Differential Geometry Forum
    Replies: 13
    Last Post: May 15th 2010, 07:34 AM
  3. example of a continuous function that is not bounded?
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: April 11th 2010, 04:01 AM
  4. Replies: 3
    Last Post: March 17th 2010, 06:12 PM
  5. continuous function on a closed and bounded set
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 10th 2009, 02:58 PM

Search Tags


/mathhelpforum @mathhelpforum