Wikipedia / Bounded function
Wikipedia / Bounded function
In mathematics, a function defined on some set with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists such that
for all. A function that is not bounded is said to be unbounded.
, then the function is said to be bounded (from) above, then the function is said to be bounded (from) below. 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 of natural numbers. Thus a sequence such that
. The set of all bounded sequences forms the sequence space.
is a bounded set.
Related notions
Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.
A bounded operator ), but has the weaker property of preserving boundedness are mapped to bounded sets . if and allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.
Examples
The sine.
except for −1 and 1, is unbounded. As .
, defined for all real , is for all .
The inverse trigonometric function is increasing for all real numbers radians
By the boundedness theorem, every continuous function, is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
which are entire are either unbounded or constant as a consequence of Liouville's theorem. must be unbounded since it is entire.
rational number and 1 for irrational number (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" is much larger than the set of continuous functions on that interval. are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)
See also
References