Bounded function

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title: "Bounded function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["complex-analysis", "real-analysis", "types-of-functions"] description: "Mathematical function whose set of values is bounded" topic_path: "general/complex-analysis" source: "https://en.wikipedia.org/wiki/Bounded_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical function whose set of values is bounded ::

[[Image:Bounded and unbounded functions.svg|right|thumb|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)|\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} 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.

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