Bounded function explained

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)|\le M for all x in X .^[1] A function that is not bounded is said to be unbounded.

If

f is real-valued and f(x) \leq A for all x in X , then the function is said to be bounded (from) above by A . If f(x) \geq B for all x in X , then the function is said to be bounded (from) below by B . 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

X is taken to be the set N of natural numbers. Thus a sequence f = (a _{0,} a _{1,} a _{2,} \ldots) is bounded if there exists a real number M such that|a _{n|\le} M for every natural number n . The set of all bounded sequences forms the sequence space l ^{infty} .

The definition of boundedness can be generalized to functions

f: X → Y taking values in a more general space Y by requiring that the image f(X) is a bounded set in Y .

Related notions

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator

T: X → Y is not a bounded function in the sense of this page's definition (unless T =0 ), but has the weaker property of preserving boundedness; bounded sets M \subseteq X are mapped to bounded sets T(M) \subseteq Y . This definition can be extended to any function f: X → Y if X and Y allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.

Examples

• The sine function

\sin: R → R is bounded since |\sin (x)| \le 1 for all x \in R .^[2]

• The function

f(x) = (x ^{2-1)} ^{-1} , defined for all real x except for −1 and 1, is unbounded. As x approaches −1 or 1, the values of this function get larger in magnitude. This function can be made bounded if one restricts its domain to be, for example, [ 2 , inf ty) or ( -inf ty, -2 ] .

• The function $f(x)=\; (x^2+1)^$, defined for all real

x , is bounded, since $|f(x)|\; \backslash le\; 1$ for all x .

• The inverse trigonometric function arctangent defined as:

y = \arctan (x) or x = \tan (y) is increasing for all real numbers x and bounded with - (\pi)/(2) < y < (\pi)/(2) radians^[3]

• By the boundedness theorem, every continuous function on a closed interval, such as

f: [ 0 , 1 ] → R , is bounded.^[4] More generally, any continuous function from a compact space into a metric space is bounded.

• All complex-valued functions

f: C → C which are entire are either unbounded or constant as a consequence of Liouville's theorem.^[5] In particular, the complex \sin: C → C must be unbounded since it is entire.

• The function

f which takes the value 0 for x rational number and 1 for x irrational number (cf. Dirichlet function) 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 larger than the set of continuous functions on that interval. Moreover, continuous functions need not be bounded; for example, the functions g: R ^{2\toR} and h: ( 0 , 1 ) ^{2\toR} defined by g(x, y) : = x + y and h(x, y) : = (1)/(x + y) are both continuous, but neither is bounded.^[6] (However, a continuous function must be bounded if its domain is both closed and bounded.)

See also

• Bounded set
• Compact support
• Local boundedness
• Uniform boundedness

Notes and References

1. Book: Jeffrey, Alan. Mathematics for Engineers and Scientists, 5th Edition. 1996-06-13. CRC Press. 978-0-412-62150-5. en.
2. Web site: The Sine and Cosine Functions. live. https://web.archive.org/web/20130202195902/https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf. 2 February 2013. 1 September 2021. math.dartmouth.edu.
3. Book: Polyanin. Andrei D.. A Concise Handbook of Mathematics, Physics, and Engineering Sciences. Chernoutsan. Alexei. 2010-10-18. CRC Press. 978-1-4398-0640-1. en.
4. Web site: Weisstein. Eric W.. Extreme Value Theorem. 2021-09-01. mathworld.wolfram.com. en.
5. Web site: Liouville theorems - Encyclopedia of Mathematics. 2021-09-01. encyclopediaofmath.org.
6. Book: Ghorpade. Sudhir R.. A Course in Multivariable Calculus and Analysis. Limaye. Balmohan V.. 2010-03-20. Springer Science & Business Media. 978-1-4419-1621-1. 56. en.