Bounded function

{{Short description|Mathematical function whose set of values is bounded}} {{More citations needed|date=September 2021}}right|thumb|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. In mathematics, a function <math>f</math> defined on some set <math>X</math> 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 <math>M</math> such that :<math>|f(x)|\le M</math> for all <math>x</math> in <math>X</math>.<ref name=":0">{{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}}</ref> A function that is ''not'' bounded is said to be '''unbounded'''.{{Citation needed|date=September 2021}}

If <math>f</math> is real-valued and <math>f(x) \leq A</math> for all <math>x</math> in <math>X</math>, then the function is said to be '''bounded (from) above''' by <math>A</math>. If <math>f(x) \geq B</math> for all <math>x</math> in <math>X</math>, then the function is said to be '''bounded (from) below''' by <math>B</math>. A real-valued function is bounded if and only if it is bounded from above and below.<ref name=":0" />{{Additional citation needed|date=September 2021}}

An important special case is a '''bounded sequence''', where ''<math>X</math>'' is taken to be the set <math>\mathbb N</math> of natural numbers. Thus a sequence <math>f = (a_0, a_1, a_2, \ldots)</math> is bounded if there exists a real number <math>M</math> such that

:<math>|a_n|\le M</math> for every natural number <math>n</math>. The set of all bounded sequences forms the sequence space <math>l^\infty</math>.{{Citation needed|date=September 2021}}

The definition of boundedness can be generalized to functions <math>f: X \rightarrow Y</math> taking values in a more general space <math>Y</math> by requiring that the image <math>f(X)</math> is a bounded set in <math>Y</math>.{{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 ''<math>T: X \rightarrow Y</math>'' is not a bounded function in the sense of this page's definition (unless <math>T=0</math>), but has the weaker property of '''preserving boundedness'''; bounded sets <math>M \subseteq X</math> are mapped to bounded sets ''<math>T(M) \subseteq Y</math>.'' This definition can be extended to any function <math>f: X \rightarrow Y</math> if ''<math>X</math>'' and ''<math>Y</math>'' 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 <math>\sin: \mathbb R \rightarrow \mathbb R</math> is bounded since <math>|\sin (x)| \le 1</math> for all <math>x \in \mathbb{R}</math>.<ref name=":0" /><ref>{{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}}</ref> * The function <math>f(x)=(x^2-1)^{-1}</math>, defined for all real <math>x</math> except for −1 and 1, is unbounded. As ''<math>x</math>'' 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, <math>[2, \infty)</math> or <math>(-\infty, -2]</math>.{{Citation needed|date= September 2021}} * The function <math display="inline">f(x)= (x^2+1)^{-1}</math>, defined for all real ''<math>x</math>'', ''is'' bounded, since <math display="inline">|f(x)| \le 1</math> for all ''<math>x</math>''.{{Citation needed|date= September 2021}} * The inverse trigonometric function arctangent defined as: <math>y= \arctan (x)</math> or <math>x = \tan (y)</math> is increasing for all real numbers ''<math>x</math>'' and bounded with <math>-\frac{\pi}{2} < y < \frac{\pi}{2}</math> radians<ref>{{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}}</ref> * By the boundedness theorem, every continuous function on a closed interval, such as <math>f: [0, 1] \rightarrow \mathbb R</math>, is bounded.<ref>{{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}}</ref> More generally, any continuous function from a compact space into a metric space is bounded.{{Citation needed|date= September 2021}} *All complex-valued functions <math>f: \mathbb C \rightarrow \mathbb C</math> which are entire are either unbounded or constant as a consequence of Liouville's theorem.<ref>{{Cite web|title=Liouville theorems - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Liouville_theorems|access-date=2021-09-01|website=encyclopediaofmath.org}}</ref> In particular, the complex <math>\sin: \mathbb C \rightarrow \mathbb C</math> must be unbounded since it is entire.{{Citation needed|date= September 2021}} * The function <math>f</math> which takes the value 0 for <math>x</math> rational number and 1 for ''<math>x</math>'' 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 <math>[0, 1]</math> 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 <math>g:\mathbb{R}^2\to\mathbb{R}</math> and <math>h: (0, 1)^2\to\mathbb{R}</math> defined by <math>g(x, y) := x + y</math> and <math>h(x, y) := \frac{1}{x+y}</math> are both continuous, but neither is bounded.<ref name=":1">{{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}}</ref> (However, a continuous function must be bounded if its domain is both closed and bounded.<ref name=":1" />)

==References== {{reflist}}

Category:Complex analysis Category:Real analysis Category:Types of functions