What Is an Example of an Essentially Bounded Function?

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In summary, an essentially bounded function is another way of saying L^\infty and any constant function is essentially bounded. It is unbounded at a sufficiently small number of places and for any epsilon>0, one can find a real number s so that the points where the function exceeds s in magnitude has measure less than epsilon. A function that is bounded is certainly essentially bounded, but the reverse is not true. An example of an essentially bounded function that is not bounded is f(x) = \begin\{ \begin{array}{cc} x & \textrm{if x is an integer} \\ 0 & \textrm{otherwise}\end{array}\right. as stated by Carl.
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Tzar
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hey
Can someone please give me an example of an essentially bounded function??

I'm a bit lost.
 
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Tzar said:
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 [tex]L^\infty[/tex] 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:

[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.

Carl
 
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Thanks Carl!
 

FAQ: What Is an Example of an Essentially Bounded Function?

What are essentially bounded functions?

Essentially bounded functions are mathematical functions that are bounded except for a finite number of points. This means that the function can take on values that are arbitrarily large or small at a finite number of points, but is otherwise bounded within a certain range.

How are essentially bounded functions different from bounded functions?

Unlike bounded functions, which are always bounded within a certain range, essentially bounded functions may have a finite number of points where the function is unbounded. This means that the function can take on values that are arbitrarily large or small at these specific points.

What is the importance of essentially bounded functions in mathematics?

Essentially bounded functions are important in various areas of mathematics, including analysis and measure theory. They allow us to define and analyze functions that may be unbounded at certain points, but still behave in a well-behaved manner overall.

How are essentially bounded functions used in real-world applications?

Essentially bounded functions are commonly used in physics, engineering, and other scientific fields to model real-world phenomena. For example, they may be used to describe the behavior of a physical system that is bounded except at a few points of interest.

How can one determine if a function is essentially bounded?

In order to determine if a function is essentially bounded, one must first define the domain of the function and identify any points where the function may be unbounded. Then, it is necessary to analyze the behavior of the function at these points and determine if they are finite or infinite. If the points are finite, the function is essentially bounded. Otherwise, it is not essentially bounded.

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