Bounded function

defined on some set

with real or complex values is called bounded if the set of its values is bounded.

In other words, there exists a real number

[1] A function that is not bounded is said to be unbounded.

, then the function is said to be bounded (from) above by

, then the function is said to be bounded (from) below by

[1][additional citation(s) needed] An important special case is a bounded sequence, where

is bounded if there exists a real number

The set of all bounded sequences forms the sequence space

[citation needed] 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

[citation needed] 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

allow for the concept of a bounded set.

Boundedness can also be determined by looking at a graph.