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Set of Points Where f Is Continuous

  1. Feb 22, 2009 #1
    The problem statement, all variables and given/known data
    Let C be the set of points where f: R --> R is continuous. Show that C may be written as the intersection of a countable collection of open sets in R.


    The attempt at a solution
    If C is empty, the result is true. If C has countably many points, say x_0, x_1, ..., then R - {x_0, x_1, ...} is the union of countable collection of open intervals, (a_i, b_i) and so we may write C as the intersection of {[a_i, b_i]}. Of course, this is not what I want, but it's the best idea I've had so far. I don't know what to do if C is uncountable. Any tips?
     
  2. jcsd
  3. Feb 22, 2009 #2
    I have another idea for the countable case: If C = {x_0, x_1, ...}, for each x_i, we may write

    x_i = intersection {(x_i - 1/n, x_i + 1/n) : n = 1, 2, ...}

    which is a countable intersection. Thus, C is the intersection of countably many intersections of countably many open intervals.
     
  4. Feb 24, 2009 #3
    Nevermind: I found a previous post about this.
     
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