Bounded function

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 $f$ 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

$|f(x)|\leq M$

$\forall$$x$ in $X$. A function that is not bounded is said to be unbounded.

If $f$ is real-valued and $f(x)\leq A,\forall x\in X$, then the function is said to be bounded (from) above by $A$. If $f(x)\geq B,\forall x\in X$, 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 $f=(a_{0},a_{1},a_{2},\dots )$ 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 $Y$.

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.

Examples

• The $\sin$ function: $R\rightarrow R$ is bounded.
• 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 and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, [2, +∞) or (-∞, −2].

• The function ${\textstyle f(x)=(x^{2}+1)^{-1}}$defined for all real $x$ is bounded.
• 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
• Every continuous function $f:$ [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's 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 bigger than the set of continuous functions on that interval.

See also

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