Bounded function

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
 * $$|f(x)|\le M$$

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


 * $$|a_n|\le M$$

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 sine function $$\sin: \mathbb R \rightarrow \mathbb R$$ is bounded since $$|\sin (x)| \le 1$$ 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 $f(x)= (x^2+1)^{-1}$, defined for all real $$x$$, is bounded, since $|f(x)| \le 1$ 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. )