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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

for all *x* in *X*.^{[1]} A function that is *not* bounded is said to be **unbounded**.^{[citation needed]}

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]}^{[additional citation(s) needed]}

An important special case is a **bounded sequence**, where *X* is taken to be the set **N** of natural numbers. Thus a sequence *f* = (*a*_{0}, *a*_{1}, *a*_{2}, ...) 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 .^{[citation needed]}

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*.^{[citation needed]}

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.^{[citation needed]}

- 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 and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, [2, ∞) or (−∞, −2].^{[citation needed]}

- The function , defined for all real
*x*,*is*bounded.^{[citation needed]} - 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.^{[citation needed]} - 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.^{[citation needed]} - 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.^{[citation needed]}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]}.)

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^{a}^{b}^{c}Jeffrey, Alan (1996-06-13).*Mathematics for Engineers and Scientists, 5th Edition*. CRC Press. ISBN 978-0-412-62150-5. **^**"The Sine and Cosine Functions" (PDF).*math.dartmouth.edu*. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021.**^**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.**^**Weisstein, Eric W. "Extreme Value Theorem".*mathworld.wolfram.com*. Retrieved 2021-09-01.**^**"Liouville theorems - Encyclopedia of Mathematics".*encyclopediaofmath.org*. Retrieved 2021-09-01.- ^
^{a}^{b}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.