# 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

for all *x* in *X*.

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*. Then the inequality above is replaced with

for some *a* in *Y*, *M*>0, and for all *x* in *X*.

## 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.