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Math Help - Metric spaces and continuous functions

  1. #1
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    Metric spaces and continuous functions

    Let (X,d) metric space. Let Y=\mathcal B\mathcal C(X,\mathbb R) be the space of every continuous and bounded functions from X to \mathbb R.

    Prove that f_1(x)=\dfrac{d(x,x_0)}{1+d(x,x_0)} and f_2(x)=d(x,x_1)-d(x,x_0) belong to Y.

    How to do this? By proving that f_1,f_2 are both continuous and bounded functions?

    Well clearly both of them are continuous. The first one is bounded by 1 and the second one can be bounded by the triangle inequality. Is that all?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Connected View Post
    Let (X,d) metric space. Let Y=\mathcal B\mathcal C(X,\mathbb R) be the space of every continuous and bounded functions from X to \mathbb R.

    Prove that f_1(x)=\dfrac{d(x,x_0)}{1+d(x,x_0)} and f_2(x)=d(x,x_1)-d(x,x_0) belong to Y.

    How to do this? By proving that f_1,f_2 are both continuous and bounded functions?

    Well clearly both of them are continuous. The first one is bounded by 1 and the second one can be bounded by the triangle inequality. Is that all?
    Indeed.
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