# Dictionary Definition

unbounded adj : seemingly boundless in amount,
number, degree, or especially extent; "unbounded enthusiasm";
"children with boundless energy"; "a limitless supply of money"
[syn: boundless,
limitless]

# User Contributed Dictionary

## English

### Adjective

- having no boundaries or limits
- The universe is finite but unbounded
- Our everlasting, unbouded love

#### Derived terms

# Extensive Definition

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 number M>0 such that

- |f(x)|\le M

Sometimes, if f(x)\le A for all x in X, then the
function is said to be bounded above by A. On the other hand, if
f(x)\ge 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 = ( a0, a1, a2, ...
) is bounded if there exists a number M > 0 such that

- |an| ≤ M

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 number M>0 such that

- d(f(x), a)\le M

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

## Examples

- The function f:R → R defined by f (x)=sin x is bounded. The sine function is no longer bounded if it is defined over the set of all complex numbers.
- The function

- f(x)=\frac

- The function

- f(x)=\frac

- Every continuous function f:[0,1] → R is bounded. This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.
- The function f which takes the value 0 for x rational number and 1 for x irrational number is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0,1] is much bigger than the set of continuous functions on that interval.

unbounded in Czech: Omezená funkce

unbounded in Esperanto: Barita funkcio

unbounded in Icelandic: Takmarkað fall

unbounded in Italian: Funzione limitata

unbounded in Hebrew: פונקציה חסומה

unbounded in Polish: Funkcja ograniczona

unbounded in Portuguese: Função
limitada

# Synonyms, Antonyms and Related Words

absolute, all-comprehensive,
all-inclusive, all-knowing, all-powerful, all-seeing, all-wise,
almighty, boundless, changeless, countless, creating, creative, endless, eternal, eternally the same,
everlasting,
exhaustless,
extending everywhere, full,
glorious, good, hallowed, highest, holy, illimitable, illimited, immeasurable, immense, immortal, immutable, incalculable, incomprehensible,
indefinite, inexhaustible, infinite, infinitely
continuous, innumerable, interminable, interminate, just, limitless, loving, luminous, majestic, making, measureless, merciful, no end of, no
strings, numinous,
omnipotent, omnipresent, omniscient, one, open, perfect, permanent, perpetual, plenary, radiant, sacred, shaping, shoreless, sovereign, sumless, supreme, termless, timeless, ubiquitous, unbound, unchanging, uncircumscribed,
unconditional,
unconditioned,
unconfined, undefined, unequivocal, unfathomable, universal, unlimited, unmeasurable, unmeasured, unnumbered, unplumbed, unqualified, unrestricted, untold, wide-open, without bound,
without end, without limit, without measure, without number,
without strings