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Continuous functions are borel

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Take f: (a,b) --> R , continuous for all x0in (a,b)
    and take (Ω = (a,b) , F = ( (a,b) [itex]\bigcap[/itex] B(R)) where B(R) is the borel sigma algebra
    Then prove f is a borel function


    3. The attempt at a solution

    I know that continuity of f means that for all x in (a,b) and all ε>0 there exists a δ>0 such that |x-x0| < δ implies |f(x)-f(x0| < ε

    And I want to show that {x in (a,b) s.t f(x) < c } is in F

    But Then I am stuck, how would I use these facts to help me ?

    Thanks in advance for any help
     
  2. jcsd
  3. May 21, 2012 #2

    micromass

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    What do you know about continuous functions?? Do you know that for a continuous function and G open that [itex]f^{-1}(G)[/itex] is open??
     
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