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Increasing funtion

  1. Nov 5, 2006 #1
    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
     
  2. jcsd
  3. Nov 5, 2006 #2

    mathwonk

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    try some simple examples, like f(x)= 0 for negative x and f(x) = 1 for non negative x.
     
  4. Nov 14, 2006 #3
    Define M = sup{f(x) | a <= x < c}, we prove

    lim f(x) = M
    x->c-

    For every e>0, there exist an element a<= d < c such that

    M-e < f(d) <= M

    Thus, for all d < x < c, we have

    M-e < f(d) <= f(x) <= M < M+e

    This means

    lim f(x) = M
    x->c-

    The second equality can be proved similarly. The two limits (left and right) are the same if the function f is continous
     
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