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 Mathematics is the study of quantity, structure, space and change. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In predicate logic, existential quantification is an attempt to formalize the notion that something (a logical predicate) is true for something, or at least one relevant thing. ...

for all x in X. In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ...

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = ( a₀, a₁, a₂, ... ) is bounded if there exists a number M > 0 such that Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... This is a page about mathematics. ...

|a_n| ≤ M

for every natural number n. The set of all bounded sequences, equipped with a vector space structure, forms a sequence space. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In functional analysis and related areas of mathematics, a sequence space is an important class of function space. ...

This definition can be extended to functions taking values in a metric space Y. Then the inequality above is replaced with In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... For the socioeconomic meaning, see social inequality. ...

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, ∞). In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

• 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 a much bigger set than the set of continuous functions.