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Math Help - Showing f is continuous

  1. #1
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    Question Showing f is continuous

    Hey,

    I am able to show how f is continuous if the domain is a discrete space. However, in this case the domain is a metric space. Therefore what would be the easiest way to show that f is continuous in this case.

    Thanks

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  2. #2
    Super Member girdav's Avatar
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    Re: Showing f is continuous

    Show that f is Lipschitz continuous thanks to the triangular inequality.
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    Re: Showing f is continuous

    Quote Originally Posted by mathshelpee View Post
    Hey,

    I am able to show how f is continuous if the domain is a discrete space. However, in this case the domain is a metric space. Therefore what would be the easiest way to show that f is continuous in this case.Thanks
    http://i53.tinypic.com/iemnv6.jpg
    Do you know the theorem: If a,~b,~\&~x_0 are points in a metric space X then \left| {d(a,x_0 ) - d(b,x_0 )} \right| \leqslant d(a,b).

    Using that you can see if \varepsilon  > 0 let \delta=\varepsilon/2 .
    Now if d(a,b)<\delta then \left| {d(a,x_0 ) - d(b,x_0 )} \right| \leqslant d(a,b)< \varepsilon .
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