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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, (∞) or (−∞, −2 ). * The function :: : defined for all real ''x'' ''is'' bounded. * Every continuous function ''f'' : (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 (1 ) is much bigger than the set of continuous functions on that interval. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bounded function」の詳細全文を読む スポンサード リンク
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