bounded function

In mathematics, a function f defined on some 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

for all x in X. An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence is bounded if there exists a number M > 0 such that

|f_n| < M

for every natural number n. This definition can be extended to functions taking values in a metric space Y. Then the inequality above is replaced with

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

for some a in Y, M>0, and for all x in X.

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{1}{x^2-1}

defined for all real x which do not equal −1 or 1 is not bounded. As x gets closer to −1 or to 1, the values of this function get larger and larger in magnitude. This function can be made bounded if one considers its domain to be for example [2, ∞).

The function

f(x)=\frac{1}{x^2+1}

defined for all real x 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 the real line is a much bigger set than the set of continuous functions.

<< Previous

Word Browser

Next >>

korotkoff sounds
common year starting on tuesday
mammary gland
kcb
prince of darkness (1987 movie)
hirschsprung's disease
supramolecular assemblies
life pornography
order of the bath
corax
tinsley viaduct

taxan
afghan constitution commission
asparukh
reverse domain hijacking
milltown
girlfriend in a coma (book)
nicolas oudinot
sandton
hovertank 3d
catacomb 3d
peter struck

acclaim entertainment
sphenoid bone
mapquest
opodeldoc
list of people who died on their birthdays
viby j
mauritius lowland forest day gecko
ecash
list of extinct languages
state senator
ljn

jean salmon macrin
mauritius upland forest day gecko
cowlitz
ultra (company)
cowlitz (tribe)
pdm
bram dijkstra
kenneth appel
cowlitz river
wolfgang haken
white river (arkansas)