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Intermediate value theorem

  1. Aug 15, 2015 #1
    Suppose that ##f## is continuous on ##[a,b]## and let ##M## be any number between ##f(a)## and ##f(b)##.
    Then, there exists a number ##c## (at least one) such that:
    ##a < c < b## and ##f(c) = M##

    Why did the author restrict ##c## to ##(a,b)## rather than ##[a,b]##? After all, ##c ∈ [a,b] ⇒ c ∈ (a,b)##.
     
  2. jcsd
  3. Aug 15, 2015 #2
    False.
     
  4. Aug 15, 2015 #3
    Oops. I just realized it's the other way around.
    The question remains; why did the author restrict ##c## to ##(a,b)## rather than ##[a,b]##?
     
  5. Aug 15, 2015 #4
    Because it gives more information.
     
  6. Aug 16, 2015 #5

    FactChecker

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    You have to be careful here. If "M any number between f(a) and f(b)" includes f(a) and f(b), then you must include the end points a & b in the conclusion ( c is in [a,b] ). Otherwise, if M is properly between f(a) and f(b) but not equal to either, then you can conclude that c is in (a,b). And that is a stronger conclusion that c in [a,b].
     
  7. Aug 20, 2015 #6
    Why is it a stronger conclusion? Is it because ##c ∈ [a,b]## implies more obvious conclusions like ##c = a ⇒ f(c) = M = f(a)##?
     
  8. Aug 20, 2015 #7

    HallsofIvy

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    It is a stronger conclusion because it restricts the region in which "c" lies- it gives more information. Using "between" to mean "strictly between" is a stronger statement.
     
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