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Essentially bounded funstions

  1. Nov 7, 2005 #1
    Can someone please give me an example of an essentially bounded function??

    I'm a bit lost.
  2. jcsd
  3. Nov 9, 2005 #2


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    "Essentially bounded" is another way of saying [tex]L^\infty[/tex] according to this link:

    So any constant function is essentially bounded.

    What you're looking for is a function which is unbounded at a sufficiently small number of places that for any epsilon>0, one can find a real number s so that the the points where the functions exceeds s in magnitude has measure less than epsilon.

    If a function is bounded, it is certainly essentially bounded. But the reverse is not true. A harder problem would be to define an essentially bounded function that is not bounded. But even that's pretty easy. For example, I think:

    [tex]f(x) = \begin\{ \begin{array}{cc} x & \textrm{if x is an integer} \\ 0 & \textrm{otherwise}\end{array}\right.[/tex]

    is an example of an unbounded function mapping R to R that is essentially bounded.

  4. Nov 17, 2005 #3
    Thanks Carl!
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