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 (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 $Y$.