"Essentially bounded" is another way of saying [tex]L^\infty[/tex] according to this link:
http://planetmath.org/encyclopedia/LpSpace.html [Broken]
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.
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