# 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

for all *x* in *X*. A function that is *not* bounded is said to be **unbounded**.

Sometimes, if *f*(*x*) ≤ *A* for all *x* in *X*, then the function is said to be **bounded above** by *A*. On the other hand, if *f*(*x*) ≥ *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 real number *M* such that

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 its distance function *d* ("distance") is less than *M*, i.e.

for all *x* in *X*.

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

## 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*except for −1 and 1 is unbounded. 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, ∞) or (−∞, −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 (cf. Dirichlet function)*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.