bounded_function

Search for "bounded function" on Ebay? Click here!

bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M < ? such that

$|f(x)|\backslash le\; M$

for all x in X. Sometimes, if $f(x)\backslash le\; A$ for all x in X, then the function is said to be bounded above by A. On the other hand, if $f(x)\backslash ge\; B$ for all x in X, then the function is said to be bounded below by B.

The concept should not be confused with that of a bounded operator.

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f =

(

a₀, a₁, a₂, ... ) is bounded if there exists a real number M < ? such that

|a_n| ? M

for every natural number n. The set of all bounded sequences, equipped with a vector space structure, forms a sequence space.

This definition can be extended to functions taking values in a metric space Y. Such a function f defined on some set X is called bounded if for some a in Y there exists a real number M < ? such that

$d(f(x),\; a)\backslash le\; M$

for all x in X.

If this is the case, there is also such an M for each other a.

Examples

• The function f:R ? R defined by f (x)=sin x is bounded. The sine function is no longer bounded if it is defined over the set of all complex numbers

• The function

$f(x)=\backslash frac\{1\}\{x^2-1\}$

defined for all real x which do not equal −1 or 1 is not bounded. As x gets closer to −1 or to 1, the values of this function get larger and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, [2, ?).

• The function

$f(x)=\backslash frac\{1\}\{x^2+1\}$

defined for all real x is bounded.

• Every continuous function f:[0,1] ? R is bounded. This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.

• The function f which takes the value 0 for x rational number and 1 for x irrational number is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0,1] is much bigger than the set of continuous functions on that interval.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "bounded_function". The list of authors you can find on this page.