Bounded function

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} 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.)

References

References

1. Jeffrey, Alan. (1996-06-13). "Mathematics for Engineers and Scientists, 5th Edition". CRC Press.
2. "The Sine and Cosine Functions".
3. (2010-10-18). "A Concise Handbook of Mathematics, Physics, and Engineering Sciences". CRC Press.
4. Weisstein, Eric W.. "Extreme Value Theorem".
5. "Liouville theorems - Encyclopedia of Mathematics".
6. (2010-03-20). "A Course in Multivariable Calculus and Analysis". Springer Science & Business Media.