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Math Help - Uniform Continuity Question

  1. #1
    Newbie FishStyx's Avatar
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    Uniform Continuity Question

    Let f: A \to B and g: B \to C be uniformly continuous. What can be said about g\circ f: A \to C?

    We have only briefly covered uniform continuity so it is unclear what I should see about g\circ f.
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  2. #2
    Super Member girdav's Avatar
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    Re: Uniform Continuity Question

    I guess A, B and C are metric spaces. Fix \varepsilon>0. Then use the uniform continuity of g on B: you will find a \delta such that |g(x)-g(y)|\leq\varepsilon if d_B(x,y)\leq \delta. Now, can you find a \delta' such that if a,a' \in A and d_A(a,a')\leq \delta' then |g \circ f(a)-g \circ f(a')|\leq \varepsilon?
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