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Math Help - help with continuous function proof

  1. #1
    Junior Member
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    help with continuous function proof

    i need help

    Let (X,d) be a metric space. Let a be element of X. Define a function f: X -> R by f(x) = d(a,x) show tat f is a continuous function.

    this is what I've done

    abs( x-y) = d(x,y) leq d(x,a) + d(a,y) by triangle inequality thm < delta = epsilon.

    now abs(f(x)-f(y)) = abs(d(a,x)-d(a,y)) leq d(x,a) + d(a,y) < epsilon

    does this make any sense???
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  2. #2
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    Here is a theorem:
    If \left\{ {a,x,y} \right\} \subseteq X then \left| {d(a,x) - d(a,y)} \right| \leqslant d(x,y). Can you prove that?
    Your problem follows at once from that theorem.
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