What Is an Example of an Essentially Bounded Function?

[Tzar]

hey
Can someone please give me an example of an essentially bounded function??

I'm a bit lost.

 

[CarlB]

hey
Can someone please give me an example of an essentially bounded function??
I'm a bit lost.

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

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:

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

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

Carl

 

[Tzar]

Thanks Carl!