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Pointwise limit.

  1. Apr 14, 2010 #1
    The set of rational numbers Q is countable, and be therefore be expressed as a countable union Q = Un>=1{rn}. Let R be a metric space with the usual distance. Now define a function f:R->R by setting

    f(x) = 1/n if x = rn and 0 if x is irrational

    (a) Show that f is continuous at each irrational point and discontinuous at each rational point.

    I did this using epsilon-delta but I thought this was pretty easy to see anyway, unless I'm completely wrong.

    (b) Show that f is the pointwise limit of a sequence of continuous functions. I.e. find a sequence of continuous functions fn such that for ever x in R then

    f(x) = lim(n->inf) fn(x)

    This is where I'm stuck and have no idea. Any pointers with this would be very helpful.
     
  2. jcsd
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