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Continuous functions proof

  1. Dec 9, 2012 #1
    Let f and g be two continuous functions on ℝ with the usual metric and let S[itex]\subset[/itex]ℝ be countable. Show that if f(x)=g(x) for all x in Sc (the complement of S), then f(x)=g(x) for all x in ℝ.

    I'm having trouble understanding how to approach this problem, can anyone give me a hint leading me in the right direction?

    Thank you.
     
  2. jcsd
  3. Dec 9, 2012 #2

    Dick

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    How about trying to show that Sc is dense in R? That would do it, yes?
     
    Last edited: Dec 9, 2012
  4. Dec 9, 2012 #3
    I was told that Sc is dense because S is countable. I'm not sure if that's a theorem, but should I just prove density using the definition or is there a simpler way?
     
  5. Dec 9, 2012 #4

    Dick

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    Use the definition. You'll have to add to that what you hopefully know about some subsets of R being uncountable.
     
    Last edited: Dec 9, 2012
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