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Darboux theorem

  1. Feb 25, 2010 #1
    Let I be an open interval in R and let f : I → R be a differentiable function.
    Let g : T → R be the function defined by g(x, y) =(f (x)−f (y))/(x-y)
    1.Prove that g(T ) ⊂ f (I) ⊂ g(T ) (The last one should be the closure of g(T), but I can't type it here)
    2. Show that f ′ (I) is an interval.

    I don't know what the closure of g(T) is, so I can't prove the second part in question 1. To show that f ′ (I) is an interval, I intend to show that it's connected, f(x) is continuous. Would that work? Or do I need to use g(t) is continuous to show that?
    Your help is greatly appreciated!
  2. jcsd
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