Misplaced Pages

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Bounded function" – news · newspapers · books · scholar · JSTOR (September 2021) (Learn how and when to remove this message)

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

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

for all ${\displaystyle x}$ in ${\displaystyle X}$. A function that is not bounded is said to be unbounded.

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.

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 ${\displaystyle l^{\infty }}$.

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 ${\displaystyle Y}$.

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.

Examples

• The sine function ${\displaystyle \sin :\mathbb {R} \rightarrow \mathbb {R} }$ is bounded since ${\displaystyle |\sin(x)|\leq 1}$ for all ${\displaystyle x\in \mathbb {R} }$.
• 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, ${\displaystyle [2,\infty )}$ ( − ∞ , − 2 ] {\displaystyle (-\infty ,-2]} .
• 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 ${\displaystyle x}$.
• 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
• By the boundedness theorem, every continuous function on a closed interval, such as ${\displaystyle f:\rightarrow \mathbb {R} }$, is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
• 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. In particular, the complex ${\displaystyle \sin :\mathbb {C} \rightarrow \mathbb {C} }$ must be unbounded since it is entire.
• 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 ${\displaystyle }$ is much larger than the set of continuous functions on that interval. 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. (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

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. 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.
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.

v t e Lp spaces
Basic concepts	Banach & Hilbert spaces L spaces Measure Lebesgue Measure space Measurable space/function Minkowski distance Sequence spaces
L spaces	Integrable function Lebesgue integration Taxicab geometry
L spaces	Bessel's Cauchy–Schwarz Euclidean distance Hilbert space Parseval's identity Polarization identity Pythagorean theorem Square-integrable function
${\displaystyle L^{\infty }}$ spaces	Bounded function Chebyshev distance Infimum and supremum Essential Uniform norm
Maps	Almost everywhere Convergence almost everywhere Convergence in measure Function space Integral transform Locally integrable function Measurable function Symmetric decreasing rearrangement
Inequalities	Babenko–Beckner Chebyshev's Clarkson's Hanner's Hausdorff–Young Hölder's Markov's Minkowski Young's convolution
Results	Marcinkiewicz interpolation theorem Plancherel theorem Riemann–Lebesgue Riesz–Fischer theorem Riesz–Thorin theorem For Lebesgue measure Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality
Applications & related	Bochner space Fourier analysis Lorentz space Probability theory Quasinorm Real analysis Sobolev space *-algebra C*-algebra Von Neumann

• Complex analysis
• Real analysis
• Types of functions