Bounded sequence

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
:$, f(x), \le M$
for all ''x'' in ''X''. A function that is ''not'' bounded is said to be unbounded.

If ''f'' is real-valued and ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be bounded (from) above by ''A''. If ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be bounded (from) below by ''B''. A real-valued function is bounded if and only if it is bounded from above and below.

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

:$, a_n, \le M$
for every natural number ''n''. The set of all bounded sequences forms the sequence space $l^\infty$.

The definition of boundedness can be generalized to functions ''f : X → Y'' taking values in a more general space ''Y'' by requiring that the image ''f(X)'' is a bounded set in ''Y''.

Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator ''T : X → Y'' is not a bounded function in the sense of this page's definition (unless ''T = 0''), but has the weaker property of preserving boundedness: Bounded sets ''M ⊆ X'' are mapped to bounded sets ''T(M) ⊆ Y.'' This definition can be extended to any function ''f'' : ''X'' → ''Y'' if ''X'' and ''Y'' allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.

Examples

* The sine function sin : R → R is bounded since $, \sin (x), \le 1$ for all $x \in \mathbf$.
* The function $f(x)=(x^2-1)^$, defined for all real ''x'' except for −1 and 1, is unbounded. As ''x'' approaches −1 or 1, the values of this function get larger in magnitude. This function can be made bounded if one restricts its domain to be, for example, , ∞) or (−∞, −2
* The function $f(x)= (x^2+1)^$, defined for all real ''x'', ''is'' bounded, since $, f(x), \le 1$ for all ''x''.
* The inverse trigonometric function arctangent defined as: ''y'' = or ''x'' = is increasing for all real numbers ''x'' and bounded with − < ''y'' < radians
* By the boundedness theorem, every continuous function on a closed interval, such as ''f'' : , 1→ R, is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
*All complex-valued functions ''f'' : C → C which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : C → C must be unbounded since it is entire.
* 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 , 1is much larger than the set of continuous functions on that interval. Moreover, continuous functions need not be bounded; for example, the functions $g:\mathbb^2\to\mathbb$ and $h: (0, 1)^2\to\mathbb$ defined by $g(x, y) := x + y$ and $h(x, y) := \frac$ are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)

See also

* Bounded set
* Compact support
* Local boundedness
* Uniform boundedness

References

{{DEFAULTSORT:Bounded Function
Real analysis
Complex analysis
Types of functions