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Inmathematics, afunction${\displaystyle f}$ defined on someset${\displaystyle X}$ withreal orcomplex values is calledbounded if the set of its values (itsimage) isbounded. In other words,there exists a real number${\displaystyle M}$ such that

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

for all${\displaystyle x}$ in${\displaystyle X}$.^[1] A function that isnot bounded is said to beunbounded.^{[citation needed]}

If${\displaystyle f}$ is real-valued and${\displaystyle f(x)\le A}$ for all${\displaystyle x}$ in${\displaystyle X}$, then the function is said to bebounded (from) above by${\displaystyle A}$. If${\displaystyle f(x)\ge B}$ for all${\displaystyle x}$ in${\displaystyle X}$, then the function is said to bebounded (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 abounded sequence, where${\displaystyle X}$ is taken to be the set${\displaystyle \mathbb{N}}$ ofnatural numbers. Thus asequence${\displaystyle f=(a_0,a_1,a_2,\dots )}$ is bounded if there exists a real number${\displaystyle M}$ such that

${\displaystyle |a_n|\le M}$

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

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

Weaker than boundedness islocal boundedness. A family of bounded functions may beuniformly bounded.

Abounded operator${\displaystyle T:X\to Y}$ is not a bounded function in the sense of this page's definition (unless${\displaystyle T=0}$), but has the weaker property ofpreserving 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${\displaystyle f:X\to Y}$ 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]}

• Thesine function${\displaystyle \sin :\mathbb{R}\to \mathbb{R}}$ is bounded since${\displaystyle |\sin (x)|\le 1}$ 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,${\displaystyle [2,\mathrm{\infty })}$ or${\displaystyle (-\mathrm{\infty },-2]}$.^{[citation needed]}
• The function$f(x)=(x^2+1{)}^{-1}$, defined for all real${\displaystyle x}$,is bounded, since$|f(x)|\le 1$ for all${\displaystyle x}$.^{[citation needed]}
• Theinverse trigonometric function arctangent defined as:${\displaystyle y=\arctan (x)}$ or${\displaystyle x=\tan (y)}$ isincreasing for all real numbers${\displaystyle x}$ and bounded withradians^[3]
• By theboundedness theorem, everycontinuous function on a closed interval, such as${\displaystyle f:[0,1]\to \mathbb{R}}$, is bounded.^[4] More generally, any continuous function from acompact space into a metric space is bounded.^{[citation needed]}
• All complex-valued functions${\displaystyle f:\mathbb{C}\to \mathbb{C}}$ which areentire are either unbounded or constant as a consequence ofLiouville's theorem.^[5] In particular, the complex${\displaystyle \sin :\mathbb{C}\to \mathbb{C}}$ 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 functiondoes 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 ofcontinuous 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])

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

• Complex analysis
• Real analysis
• Types of functions

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