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Math Help - uniform continuity problems

  1. #1
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    uniform continuity problems

    I need help solving the following:
    Assume f and g are uniformly continuous on D.

    First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

    I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

    Also, if g(x)\neq 0 for any x in D and D closed and bounded, then \frac{f}{g} is uniformly continuous.
    Any help on these is greatly appreciated thanks!
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by dannyboycurtis View Post
    I need help solving the following:
    Assume f and g are uniformly continuous on D.

    First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

    I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

    Also, if g(x)\neq 0 for any x in D and D closed and bounded, then \frac{f}{g} is uniformly continuous.
    Any help on these is greatly appreciated thanks!
    This inequality should help you:

    |f(x)g(x)-f(y)g(y)|\leq|f(x)||g(x)-g(y)|+|g(y)||f(x)-f(y)|

    For the second one, use the fact that D is closed and bounded and that f and g are u.c. to deduce that they are bounded and then it's reduced to part 1.
    Last edited by redsoxfan325; November 2nd 2009 at 09:32 PM. Reason: Fixed typo
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