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Thread: Distance function: continuous?

  1. #1
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    Distance function: continuous?

    Let (X,d) be a metric space. Prove that the distance function
    $\displaystyle d: X \times X \rightarrow \mathbb{R} $ is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Let (X,d) be a metric space. Prove that the distance function
    $\displaystyle d: X \times X \rightarrow \mathbb{R} $ is continuous, assuming that XxX has the product topology that results from each copy of X having the topology induced by d.
    Remember, a function $\displaystyle f:R \rightarrow R $ is said to be continuous if, at the point $\displaystyle a \in R$, if given $\displaystyle \epsilon > 0 $, there is a $\displaystyle \delta > 0 $ such that

    $\displaystyle |f(x) - f(a)| < \epsilon $

    whenever:

    $\displaystyle |x-a| < \delta $
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