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A function without a maximum and a minimum

  1. Mar 11, 2012 #1
    1. The problem statement, all variables and given/known data

    Give an example of a non-continuous function on [0,1] that has no maximum and no minimum.

    2. Relevant equations

    Well, a continuous function on a non-empty compact set will have a maximum and a minimum, so I guess this is why we need a non-continuous function.

    3. The attempt at a solution

    Does this work?

    f(x) =

    1/2 if x is a rational number
    x if x is an irrational number.

    Does this work? Breaking this up into 1/2s (if rational) and x's (if irrational) hopefully makes it non-continuous, and I suppose there is always some larger number that could be formed as we approach 1 (so no maximum). What do you think?
     
  2. jcsd
  3. Mar 11, 2012 #2
    I would agree.
     
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