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Math Help - Uniformly Continuous f and g functions

  1. #1
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    Uniformly Continuous f and g functions

    I've been trying to figure out a couple of questions, with no luck. Any help would be really appreciated.

    1. Let f and g be real valued functions that are uniformly continuous on D. Prove that f + g is uniformly continuous on D.

    2. Let f and g be real valued functions that are uniformly continuous on D, and suppose that g(x) does not equal 0 for all x is an element of D.

    a. Find an example to show that the function f/g need not be uniformly continuous on D.

    b. Prove that if D is compact, then f/g must be uniformly continous on D.


    Again, thank you to anyone for any help.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by QPerfection View Post
    I've been trying to figure out a couple of questions, with no luck. Any help would be really appreciated.

    1. Let f and g be real valued functions that are uniformly continuous on D. Prove that f + g is uniformly continuous on D.
    Just do it.

    2. Let f and g be real valued functions that are uniformly continuous on D, and suppose that g(x) does not equal 0 for all x is an element of D.

    a. Find an example to show that the function f/g need not be uniformly continuous on D.
    Hint:

    Spoiler:
    Think about D=(0,1),f(x)=x,g(x)=\cdots


    b. Prove that if D is compact, then f/g must be uniformly continous on D.
    What's that one theorem about continuous functions on compact subspaces of \mathbb{R}?
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