Encyclopedia

Bounded function

Mathematical function whose set of values is bounded

Wikipedia contributorsAdapted for Fathomly · View authors

2 min read
Source updated August 13, 2025

In mathematics, a function $f$ defined on some set $X$ 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 $M$ such that

$|f(x)|\leq M$

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

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.

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

$|a_{n}|\leq M$

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

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 $Y$ .

02Examples

The sine function $\sin :\mathbb {R} \rightarrow \mathbb {R}$ is bounded since $|\sin(x)|\leq 1$ for all $x\in \mathbb {R}$ .
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 $(-\infty ,-2]$ .
The function ${\textstyle f(x)=(x^{2}+1)^{-1}$ , defined for all real $x$ , is bounded, since ${\textstyle |f(x)|\leq 1$ for all $x$ .
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
By the boundedness theorem, every continuous function on a closed interval, such as $f:[0,1]\rightarrow \mathbb {R}$ , is bounded. More generally, any continuous function from a compact space into a metric space is bounded.
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. In particular, the complex $\sin :\mathbb {C} \rightarrow \mathbb {C}$ must be unbounded since it is entire.
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. 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. (However, a continuous function must be bounded if its domain is both closed and bounded.)

Watch videos about Bounded functionExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Bounded function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Bounded and unbounded functions.svg by Oleg Alexandrov, Public domain

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Bounded function

02Examples

Sources and credits

Related topics

Bounded set

Support (mathematics)

Local boundedness

Bounded function

01Related notions

02Examples

Sources and credits

Related topics

Bounded set

Support (mathematics)

Local boundedness