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 number "M">0 such that :$|f(x)|le\; M$
for all "x" in "X".

Sometimes, if $f(x)le\; A$ for all "x" in "X", then the function is said to be bounded above by "A". On the other hand, if $f(x)ge\; B$ 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" = ("a"₀, "a"₁, "a"₂, ... ) is bounded if there exists a number "M" > 0 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 number "M">0 such that

:$d(f(x),\; a)le\; M$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 :$f(x)=frac\{1\}\{x^2-1\}$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 :$f(x)=frac\{1\}\{x^2+1\}$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.