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Continuous function

  1. Oct 20, 2008 #1
    I have an assignment question

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

    I'm not sure what this function looks like. Is f(x) = sqrt(a^2+x^2) and if it is I need abs(x-a) < delta?? I'm confused.
  2. jcsd
  3. Oct 20, 2008 #2
    That would be one possibility, but the problem, as written, is not specific to a particular metric. You need to show the result for an arbitrary metric. Note that all metrics have the following 4 properties:

    1) [itex] d(x,y) \geq 0[/itex]
    2) [itex] d(x,y) = 0 [/itex] if and only if [itex]x = y[/itex]
    3) [itex] d(x,y) = d(y,x)[/itex]
    4) [itex] d(x,z) \leq d(x,y) + d(y,z) [/itex]

    What you want to show is that if [itex]|f(x) - f(y)| < \epsilon[/itex], then there exists some [itex]\delta[/itex] such that [itex]d(x,y) < \delta[/itex]. Given the construction of [itex]f(x)[/itex] used here, I would expect property 4 (aka the Triangle Inequality) to be useful here.

    Sorry - LaTeX rendering seems to be broken. Please refer to the underlying text in the meantime (hit quote and you will see it).
  4. Oct 21, 2008 #3
    I think I have it now, is delta = epsilon? from the triangle inequality? if d(a,x),d(a,x_o) < epsilon/2
    Last edited: Oct 21, 2008
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