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Math Help - Help with proving incresing function on an interval

  1. #1
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    Help with proving incresing function on an interval

    A proper open subinterval of the open interval (a, b) subset of R is an open
    interval (c, d) such that (c, d) is subset but not equal to (a, b) (i.e. a < c < d < b).

    Suppose that f : (a, b) --> R is increasing on every proper open subinterval of
    (a, b). Prove that f is increasing on (a, b).
    Last edited by 450081592; November 15th 2009 at 12:42 AM.
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  2. #2
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    Quote Originally Posted by 450081592 View Post
    A proper open subinterval of the open interval (a, b) subset of R is an open
    interval (c, d) such that (c, d) not subset of (a, b) (i.e. a < c < d < b).
    That "not" is incorrect. You mean to say (c,d) is a subset of (a,b)

    Suppose that f : (a, b) --> R is increasing on every proper open subinterval of
    (a, b). Prove that f is increasing on (a, b).
    If you allowed (a,c) or (c, b) where a< c< b as "proper" subintervals, that would be easy.

    If not, I think a proof by contradiction would be best. If f is not increasing on (a,b), there exist a< c< d< b with f(c)> f(d). Now look at the interval (c,d).
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    Quote Originally Posted by HallsofIvy View Post
    That "not" is incorrect. You mean to say (c,d) is a subset of (a,b)


    If you allowed (a,c) or (c, b) where a< c< b as "proper" subintervals, that would be easy.

    If not, I think a proof by contradiction would be best. If f is not increasing on (a,b), there exist a< c< d< b with f(c)> f(d). Now look at the interval (c,d).

    ya sorry I meant that (c,d) is subset but not equal to (a,b) i.e a proper subset
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