TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
''f'' defined on some set ''X'' with
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

values is called bounded if the set of its values is bounded. In other words,
there exists In predicate logic, an existential quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by ...
a real number ''M'' such that :$, f\left(x\right), \le M$
for all In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alg ...
''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 number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s. Thus a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

''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 In functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functiona ...
$l^\infty$. 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 An artist's impression of a bounded set (top) and of an unbounded set (bottom). The set at the bottom continues forever towards the right. :''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology) In topology ...
in ''Y''.

# Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be
uniformly bounded In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. A
bounded operator In functional analysis, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps Bounded set (topological vector space), bounded subsets of to bounded subsets of . If and are normed vector sp ...
''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 function sin : R → R is bounded. * The function $f\left(x\right)=\left(x^2-1\right)^$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 and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, , ∞) or (−∞, −2 * The function $f(x)= (x^2+1)^$defined for all real ''x'' ''is'' bounded. * The
inverse trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
arctangent defined as: ''y'' = or ''x'' = is
increasing Image:Monotonicity example3.png, Figure 3. A function that is not monotonic In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that pres ...
for all real numbers ''x'' and bounded with − < ''y'' <
radians The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric s ...

* Every
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''f'' :
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
→ R is bounded. More generally, any continuous function from a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
into a metric space is bounded. *All complex-valued functions ''f'' : C → C which are
entire *In philately, see Cover (philately), Cover *In mathematics, see Entire function *In animal fancy and animal husbandry, entire (animal), entire indicates that an animal has not been desexed, that is, spayed or neutered *In botany, an edge (such as o ...
are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : C → C must be unbounded since it's entire. * The function ''f'' which takes the value 0 for ''x''
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
and 1 for ''x''
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
(cf.
Dirichlet function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
) ''is'' bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
is much bigger than the set of
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s on that interval.