What is "bounded function"

n. (context mathematics English) Any function whose values remain bounded by some constant.

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

∣f(x)∣ ≤ M
for all x in X. A function that is not bounded is said to be unbounded.

Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B.

The concept should not be confused with that of a bounded operator.

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = (a, a, a, ...) is bounded if there exists a real number M such that

∣a∣ ≤ M
for every natural number n. The set of all bounded sequences, equipped with a vector space structure, forms a sequence space.

This definition can be extended to functions taking values in a metric space Y. Such a function f defined on some set X is called bounded if for some a in Y there exists a real number M such that its distance function d ("distance") is less than M, i.e.

d(f(x), a) ≤ M

for all x in X.

If this is the case, there is also such an M for each other a, by the triangle inequality.