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Math Help - increasing functions

  1. #1
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    Question increasing functions

    A function f : A ->R is increasing if f(x) <= f(y) for every x, y in A such that x <= y.

    Suppose that f : [a, b] -> R is increasing and that a < c < b.

    i want to shat that :
    lim f(x) = sup{f(x) | a <= x < c} and
    x->c-

    limf(x) = inf{f(x) | c < x <= b}.
    x->c+

    and whether these limits are the same?

    can anyone help with this...any guidance..or starting me off thankz
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  2. #2
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    Quote Originally Posted by dopi View Post
    and whether these limits are the same?
    I do not think so, for the functions can be discontinous.
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  3. #3
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    Question Is this what i need to use

    The set L={f(x): x in[a,b]]

    Then L is bounded above by . f(c)

    Thus L has a least upper bound, say B= sup{L}
    So, . epsilon >0 => (there exists f(t) in L)[B-epsilon<f(t)<= B]
    this is the definition of lim f(x( as x->c ...this is what i got from my class notes...is this what i could use to prove solve the question i posted before...can anyone show me please thankz
    Last edited by dopi; November 5th 2006 at 10:57 AM. Reason: forgot tiltle
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  4. #4
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    Quote Originally Posted by dopi View Post
    The set L={f(x): x in[a,b]]

    Then L is bounded above by . f(c)

    Thus L has a least upper bound, say B= sup{L}
    So, . epsilon >0 => (there exists f(t) in L)[B-epsilon<f(t)<= B]
    this is the definition of lim f(x( as x->c ...this is what i got from my class notes...is this what i could use to prove solve the question i posted before...can anyone show me please thankz
    I think you should say,
    \{ f(x)|x \in [a,c)\} is bounded by f(c) because the function is increasing. Then by Bolzano-Weirestass theorem \lim_{x\to c^-}f(x) exist, also it is the least upper bound that is the supremum of the sequence. Similar argument for the other half.
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