Bounded function
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In mathematics, a function defined on some set with real oder complex values is called bounded if the set of its values is bounded. In other words, there exists a real number such that
in . A function that is not bounded is said to be unbounded.
If is real-valued and , then the function is said to be bounded (from) above by . If , then the function is said to be bounded (from) below by . 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 is taken to be the set of natural numbers. Thus a sequence is bounded if there exists a real number such that
for every natural number . The set of all bounded sequences forms the sequence space .
The definition of boundedness can be generalized to functions taking values in a more general space by requiring that the image is a bounded set in .
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 ), but has the weaker property of preserving boundedness: Bounded sets are mapped to bounded sets . This definition can be extended to any function if and allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.
Examples
- The function: is bounded.
- The function defined for all real except for −1 and 1 is unbounded. As approaches −1 or 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, +∞) oder (-∞, −2].
- The function defined for all real is bounded.
- The inverse trigonometric function arctangent defined as: oder is increasing for all real numbers and bounded with radians
- Every continuous function [0, 1] is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
- All complex-valued functions which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex : must be unbounded since it's entire.
- The function which takes the value 0 for rational number and 1 for 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 [0, 1] is much bigger than the set of continuous functions on that interval.