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In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. ==Locally bounded function== A real-valued or complex-valued function ''f'' defined on some topological space ''X'' is called locally bounded if for any ''x''0 in ''X'' there exists a neighborhood ''A'' of ''x''0 such that ''f'' (''A'') is a bounded set, that is, for some number ''M''>0 one has : for all ''x'' in ''A''. That is to say, for each ''x'' one can find a constant, depending on ''x'', which is larger than all the values of the function in the neighborhood of ''x''. Compare this with a bounded function, for which the constant does not depend on ''x''. Obviously, if a function is bounded then it is locally bounded. The converse is not true in general. This definition can be extended to the case when ''f'' takes values in some metric space. Then the inequality above needs to be replaced with : for all ''x'' in ''A'', where ''d'' is the distance function in the metric space, and ''a'' is some point in the metric space. The choice of ''a'' does not affect the definition. Choosing a different ''a'' will at most increase the constant ''M'' for which this inequality is true. 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「Local boundedness」の詳細全文を読む スポンサード リンク
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