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Monotonically decreasing function

  1. Jul 19, 2007 #1


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    1. The problem statement, all variables and given/known data
    True or false:
    1) If f:R->R is a monotonically decreasing function then every discontinues point of f has finate right and left limits which are unequal.
    2) I is some finate segment where a<b and a,b in R. If every continues function defined in I has a maximum and a minimum then I is a closed (in other words [a,b]) segment.

    2. Relevant equations

    3. The attempt at a solution

    1) I think this is true but how can I prove it? Can someone give me a push in the right direction?
    2)True: f(x)=x is continues and defined in any segment but it only has a min and max in an closed one. Is that right?

  2. jcsd
  3. Jul 19, 2007 #2


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    2) looks fine. For 1) try thinking of it this way. Let c be the discontinuous point. For the left limit consider the sequence f(c-1/n) for integers n>=1. The sequence is decreasing and bounded below (by f(c)) so it has a limit L. Can you show R is the left hand limit?
    Last edited: Jul 19, 2007
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