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

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.

for all x in X.[1] A function that is not bounded is said to be unbounded.

If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ 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.[1]

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = (a0, a1, a2, ...) is bounded if there exists a real number M such that

for every natural number n. The set of all bounded sequences forms the sequence space .

The definition of boundedness can be generalized to functions f : X → Y taking values in a more general space Y by requiring that the image f(X) is a bounded set in Y.

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator T : X → 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 ⊆ X are mapped to bounded sets T(M) ⊆ Y. This definition can be extended to any function f : X → 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 : R → R is bounded since for all .[1][2]
  • The function , 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, ∞) or (−∞, −2].
  • The function , defined for all real x, is bounded, since 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 −π/2 < y < π/2 radians[3]
  • By the boundedness theorem, every continuous function on a closed interval, such as f : [0, 1] → R, is bounded.[4] More generally, any continuous function from a compact space into a metric space is bounded.
  • All complex-valued functions f : C → C which are entire are either unbounded or constant as a consequence of Liouville's theorem.[5] In particular, the complex sin : C → 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 and defined by and are both continuous, but neither is bounded.[6] (However, a continuous function must be bounded if its domain is both closed and bounded.[6])

See also

References
  1. Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5.
  2. "The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021.
  3. Polyanin, Andrei D.; Chernoutsan, Alexei (2010-10-18). A Concise Handbook of Mathematics, Physics, and Engineering Sciences. CRC Press. ISBN 978-1-4398-0640-1.
  4. Weisstein, Eric W. "Extreme Value Theorem". mathworld.wolfram.com. Retrieved 2021-09-01.
  5. "Liouville theorems - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-09-01.
  6. Ghorpade, Sudhir R.; Limaye, Balmohan V. (2010-03-20). A Course in Multivariable Calculus and Analysis. Springer Science & Business Media. p. 56. ISBN 978-1-4419-1621-1.
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