Bounded function

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A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not. Bounded and unbounded functions.svg
A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.

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

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for all x in X. A function that is not bounded is said to be unbounded.

If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ 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 N of natural numbers. Thus a sequence f = (a0, a1, a2, ...) 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 .

The definition of boundedness can be generalized to functions f : X → Y taking values in a more general space Y by requiring that the image f(X) is a bounded set in Y.

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator T : X → Y 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 M ⊆ X are mapped to bounded sets T(M) ⊆ Y. This definition can be extended to any function f : XY if X and Y allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.

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