Bounded function

{{Short description|A mathematical function the set of whose values is bounded}}

{{More citations needed|date=September 2021}}Image:Bounded and unbounded functions.svg

In mathematics, a function $f$ 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 $X$ .{{Cite book|last=Jeffrey|first=Alan|url=https://books.google.com/books?id=jMUbUCUOaeQC&dq=%22Bounded+function%22&pg=PA66|title=Mathematics for Engineers and Scientists, 5th Edition|date=1996-06-13|publisher=CRC Press|isbn=978-0-412-62150-5|language=en}} A function that is not bounded is said to be unbounded.{{Citation needed|date=September 2021}}

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.{{Additional citation needed|date=September 2021}}

An important special case is a bounded sequence, where $X$ is taken to be the set $\mathbb N$ 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 $l^\infty$ .{{Citation needed|date=September 2021}}

The definition of boundedness can be generalized to functions $f: X \rightarrow Y$ taking values in a more general space $Y$ by requiring that the image $f(X)$ is a bounded set in $Y$ .{{Citation needed|date= September 2021}}

Related notions

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

A bounded operator $T: X \rightarrow Y$ 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 $f: X \rightarrow Y$ if $X$ and $Y$ allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.{{Citation needed|date= September 2021}}

Examples

The sine function $\sin: \mathbb R \rightarrow \mathbb R$ is bounded since $|\sin (x)| \le 1$ for all $x \in \mathbb{R}$ .{{Cite web|title=The Sine and Cosine Functions|url=https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf|url-status=live|archive-url=https://web.archive.org/web/20130202195902/https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf|archive-date=2 February 2013|access-date=1 September 2021|website=math.dartmouth.edu}}
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|date= September 2021}}
The function $f(x)= (x^2+1)^{-1}$ , defined for all real $x$ , is bounded, since $|f(x)| \le 1$ for all $x$ .{{Citation needed|date= September 2021}}
The inverse trigonometric function arctangent defined as: $y= \arctan (x)$ or $x = \tan (y)$ is increasing for all real numbers $x$ and bounded with $-\frac{\pi}{2} < y < \frac{\pi}{2}$ radians{{Cite book|last1=Polyanin|first1=Andrei D.|url=https://books.google.com/books?id=ejzScufwDRUC&dq=arctangent+bounded&pg=PA27|title=A Concise Handbook of Mathematics, Physics, and Engineering Sciences|last2=Chernoutsan|first2=Alexei|date=2010-10-18|publisher=CRC Press|isbn=978-1-4398-0640-1|language=en}}
By the boundedness theorem, every continuous function on a closed interval, such as $f: [0, 1] \rightarrow \mathbb R$ , is bounded.{{Cite web|last=Weisstein|first=Eric W.|title=Extreme Value Theorem|url=https://mathworld.wolfram.com/ExtremeValueTheorem.html|access-date=2021-09-01|website=mathworld.wolfram.com|language=en}} More generally, any continuous function from a compact space into a metric space is bounded.{{Citation needed|date= September 2021}}
All complex-valued functions $f: \mathbb C \rightarrow \mathbb C$ which are entire are either unbounded or constant as a consequence of Liouville's theorem.{{Cite web|title=Liouville theorems - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Liouville_theorems|access-date=2021-09-01|website=encyclopediaofmath.org}} In particular, the complex $\sin: \mathbb C \rightarrow \mathbb C$ must be unbounded since it is entire.{{Citation needed|date= September 2021}}
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|date= September 2021}} Moreover, continuous functions need not be bounded; for example, the functions $g:\mathbb{R}^2\to\mathbb{R}$ and $h: (0, 1)^2\to\mathbb{R}$ defined by $g(x, y) := x + y$ and $h(x, y) := \frac{1}{x+y}$ are both continuous, but neither is bounded.{{Cite book|last1=Ghorpade|first1=Sudhir R.|url=https://books.google.com/books?id=JVFJAAAAQBAJ&q=%22Bounded+function%22|title=A Course in Multivariable Calculus and Analysis|last2=Limaye|first2=Balmohan V.|date=2010-03-20|publisher=Springer Science & Business Media|isbn=978-1-4419-1621-1|pages=56|language=en}} (However, a continuous function must be bounded if its domain is both closed and bounded.)

References

Category:Complex analysis

Category:Real analysis

Category:Types of functions

Bounded function

Related notions

Examples

See also

References