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Functions attaining max value => domain compact

  1. Nov 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that if every continuous real-valued function on a set X attains a maximum value, then X must be compact.


    2. Relevant equations
    None really.


    3. The attempt at a solution
    None really, not sure where to start. I know that if a space is compact, every function attains it's minimum and maximum values, and I was thinking that I could show that if the space is not compact, then there is some continuous real-valued function that cannot attain a max value. I'm not really sure where to begin, though. Suggestions?
     
  2. jcsd
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