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Thread: Uniform Continuity Question

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

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

    We have only briefly covered uniform continuity so it is unclear what I should see about $\displaystyle g\circ f$.
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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 $\displaystyle \varepsilon>0$. Then use the uniform continuity of $\displaystyle g$ on $\displaystyle B$: you will find a $\displaystyle \delta$ such that $\displaystyle |g(x)-g(y)|\leq\varepsilon$ if $\displaystyle d_B(x,y)\leq \delta$. Now, can you find a $\displaystyle \delta'$ such that if $\displaystyle a,a' \in A$ and $\displaystyle d_A(a,a')\leq \delta'$ then $\displaystyle |g \circ f(a)-g \circ f(a')|\leq \varepsilon$?
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