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 (its image) is bounded. In other words, there exists a real number $M$ such that

| f (x) | \leq M

for all $x$ in $X$ .^[1] A function that is not bounded is said to be unbounded.^{[citation needed]}

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.^[1]^{[additional citation(s) needed]}

An important special case is a bounded sequence, where $X$ is taken to be the set $ℕ$ of natural numbers. Thus a sequence $f = (a_{0}, a_{1}, a_{2}, \dots)$ is bounded if there exists a real number $M$ such that

| a_{n} | \leq M

for every natural number $n$ . The set of all bounded sequences forms the sequence space $l^{\infty}$ .^{[citation needed]}

The definition of boundedness can be generalized to functions $f : X \to Y$ taking values in a more general space $Y$ by requiring that the image $f (X)$ is a bounded set in $Y$ .^{[citation needed]}

Related notions

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

A bounded operator $T : X \to Y$ 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 $f : X \to Y$ if $X$ and $Y$ allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.^{[citation needed]}

Examples

The sine function $\sin : ℝ \to ℝ$ is bounded since $| \sin (x) | \leq 1$ for all $x \in ℝ$ .^[1]^[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, $[2, \infty)$ or $(- \infty, - 2]$ .^{[citation needed]}
The function $f (x) = (x^{2} + 1)^{- 1}$ , defined for all real $x$ , is bounded, since $| f (x) | \leq 1$ for all $x$ .^{[citation needed]}
The inverse trigonometric function arctangent defined as: $y = \arctan (x)$ or $x = \tan (y)$ is increasing for all real numbers $x$ and bounded with $- \frac{π}{2} < y < \frac{π}{2}$ radians^[3]
By the boundedness theorem, every continuous function on a closed interval, such as $f : [0, 1] \to ℝ$ , is bounded.^[4] More generally, any continuous function from a compact space into a metric space is bounded.^{[citation needed]}
All complex-valued functions $f : ℂ \to ℂ$ which are entire are either unbounded or constant as a consequence of Liouville's theorem.^[5] In particular, the complex $\sin : ℂ \to ℂ$ must be unbounded since it is entire.^{[citation needed]}
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.^{[citation needed]} Moreover, continuous functions need not be bounded; for example, the functions $g : ℝ^{2} \to ℝ$ and $h : (0, 1)^{2} \to ℝ$ defined by $g (x, y) := x + y$ and $h (x, y) := \frac{1}{x + y}$ are both continuous, but neither is bounded.^[6] (However, a continuous function must be bounded if its domain is both closed and bounded.^[6])

References

1 2 3 Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5.
↑ "The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021.
↑ Polyanin, Andrei D.; Chernoutsan, Alexei (2010-10-18). A Concise Handbook of Mathematics, Physics, and Engineering Sciences. CRC Press. ISBN 978-1-4398-0640-1.
↑ Weisstein, Eric W. "Extreme Value Theorem". mathworld.wolfram.com. Retrieved 2021-09-01.
↑ "Liouville theorems - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-09-01.
1 2 Ghorpade, Sudhir R.; Limaye, Balmohan V. (2010-03-20). A Course in Multivariable Calculus and Analysis. Springer Science & Business Media. p. 56. ISBN 978-1-4419-1621-1.

[:0-1] 1 2 3 Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5.

[2] "The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021.

[3] Polyanin, Andrei D.; Chernoutsan, Alexei (2010-10-18). A Concise Handbook of Mathematics, Physics, and Engineering Sciences. CRC Press. ISBN 978-1-4398-0640-1.

[4] Weisstein, Eric W. "Extreme Value Theorem". mathworld.wolfram.com. Retrieved 2021-09-01.

[5] "Liouville theorems - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-09-01.

[:1-6] 1 2 Ghorpade, Sudhir R.; Limaye, Balmohan V. (2010-03-20). A Course in Multivariable Calculus and Analysis. Springer Science & Business Media. p. 56. ISBN 978-1-4419-1621-1.

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Related notions

Examples

See also

References