Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Increasing funtion
  1. Eigen Funtions (Replies: 2)

Loading...