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Math Help - Continuity characterization help

  1. #1
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    Continuity characterization help

    Hello guys!

    Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map. Suppose that for each a>0 there exists b>0 such that for all x in X we have:


    B(f(x), b) is contained in closure( f(B(x,a))).



    Here B(f(x),b) represents the open ball with centre f(x) and radius b.
    Similarly B(x,a) represents the open ball with centre x and radius a.



    Prove that for all x in X and for every c > a :

    B(f(x), b) is contained in f(B(x,c)).


    I tried to use the characterization of continuity in terms of the closure but got stuck.

    Any ideas?
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  2. #2
    Senior Member Sampras's Avatar
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    Quote Originally Posted by Carl140 View Post
    Hello guys!

    Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map. Suppose that for each a>0 there exists b>0 such that for all x in X we have:


    B(f(x), b) is contained in closure( f(B(x,a))).



    Here B(f(x),b) represents the open ball with centre f(x) and radius b.
    Similarly B(x,a) represents the open ball with centre x and radius a.



    Prove that for all x in X and for every c > a :

    B(f(x), b) is contained in f(B(x,c)).


    I tried to use the characterization of continuity in terms of the closure but got stuck.

    Any ideas?
    Since  f is continuous, for all  E \subseteq X ,  f(\overline{E}) \subseteq \overline{f(E)} . Now we know that  B(f(x),b) \subseteq \overline{f(B(x,a))} . We want to show that  B(f(x),b) \subseteq f(B(x,c)) for every  c>a . I think it boils down to showing that  \overline{f(B(x,a))} \subseteq f(B(x,c)) .
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  3. #3
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    Right, I have tried to do that, but got stuck. Can you please help a little bit?
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