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. If f is real-valued and f(x) \leq A for all x in X, then the function is said to be bounded (from) above by A. If f(x) \geq 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 \mathbb N of natural numbers. Thus a sequence is bounded if there exists a real number M such that for every natural number n. The set of all bounded sequences forms the sequence space l^\infty. The definition of boundedness can be generalized to functions taking values in a more general space Y by requiring that the image f(X) is a bounded set in Y.

Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded. A bounded operator **' 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 are mapped to bounded sets . This definition can be extended to any function if X and Y'' allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.

Examples