Bounded function

In mathematics, a function ${\displaystyle f}$ defined on some set ${\displaystyle X}$ with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ${\displaystyle M}$ such that

A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.

${\displaystyle |f(x)|\leq M}$

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

If ${\displaystyle f}$ is real-valued and ${\displaystyle f(x)\leq A}$ for all ${\displaystyle x}$ in ${\displaystyle X}$, then the function is said to be bounded (from) above by ${\displaystyle A}$. If ${\displaystyle f(x)\geq B}$ for all ${\displaystyle x}$ in ${\displaystyle X}$, then the function is said to be bounded (from) below by ${\displaystyle B}$. 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 ${\displaystyle X}$ is taken to be the set ${\displaystyle \mathbb {N} }$ of natural numbers. Thus a sequence ${\displaystyle f=(a_{0},a_{1},a_{2},\ldots )}$ is bounded if there exists a real number ${\displaystyle M}$ such that

${\displaystyle |a_{n}|\leq M}$

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

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

Related notions

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

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

Examples

• The sine function ${\displaystyle \sin :\mathbb {R} \rightarrow \mathbb {R} }$  is bounded since ${\displaystyle |\sin(x)|\leq 1}$  for all ${\displaystyle x\in \mathbb {R} }$ .^[1][2]
• The function ${\displaystyle f(x)=(x^{2}-1)^{-1}}$ , defined for all real ${\displaystyle x}$  except for −1 and 1, is unbounded. As ${\displaystyle x}$  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, ${\displaystyle [2,\infty )}$  or ${\displaystyle (-\infty ,-2]}$ .^{[citation needed]}
• The function ${\textstyle f(x)=(x^{2}+1)^{-1}}$ , defined for all real ${\displaystyle x}$ , is bounded, since ${\textstyle |f(x)|\leq 1}$  for all ${\displaystyle x}$ .^{[citation needed]}
• The inverse trigonometric function arctangent defined as: ${\displaystyle y=\arctan(x)}$  or ${\displaystyle x=\tan(y)}$  is increasing for all real numbers ${\displaystyle x}$  and bounded with ${\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}$  radians^[3]
• By the boundedness theorem, every continuous function on a closed interval, such as ${\displaystyle f:[0,1]\rightarrow \mathbb {R} }$ , is bounded.^[4] More generally, any continuous function from a compact space into a metric space is bounded.^{[citation needed]}
• All complex-valued functions ${\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }$  which are entire are either unbounded or constant as a consequence of Liouville's theorem.^[5] In particular, the complex ${\displaystyle \sin :\mathbb {C} \rightarrow \mathbb {C} }$  must be unbounded since it is entire.^{[citation needed]}
• The function ${\displaystyle f}$  which takes the value 0 for ${\displaystyle x}$  rational number and 1 for ${\displaystyle x}$  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 ${\displaystyle [0,1]}$  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 ${\displaystyle g:\mathbb {R} ^{2}\to \mathbb {R} }$  and ${\displaystyle h:(0,1)^{2}\to \mathbb {R} }$  defined by ${\displaystyle g(x,y):=x+y}$  and ${\displaystyle h(x,y):={\frac {1}{x+y}}}$  are both continuous, but neither is bounded.^[6] (However, a continuous function must be bounded if its domain is both closed and bounded.^[6])

See also

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

References

1. 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. (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.
6. 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.