Bounded Function
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.
Sometimes, if for all x in X, then the function is said to be bounded above by A. On the other hand, if 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 = ( 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, 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.
for all x in X.
If this is the case, there is also such an M for each other a, by the triangle inequality.
Read more about Bounded Function: Examples
Famous quotes containing the words bounded and/or function:
“Me, what’s that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)
“As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
—Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)