This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2009) 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. 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 = (a0, a1, a2, ...) 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 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.Source: WikipediaText from this biography licensed under creative commons license