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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 ≤ A for all x in X, then the function is said to be bounded above by A. If f ≥ B for all x in X, then the function is said to be bounded 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 = is bounded if there exists a real number M such that
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