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''.

Contents

Examples

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
:
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
:
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.