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Bounded function
Mathematical function

In mathematics, a function f {\displaystyle f} defined on some set X {\displaystyle 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 {\displaystyle M} such that

| f ( x ) | ≤ M {\displaystyle |f(x)|\leq M}

for all x {\displaystyle x} in X {\displaystyle X} . A function that is not bounded is said to be unbounded.

If f {\displaystyle f} is real-valued and f ( x ) ≤ A {\displaystyle f(x)\leq A} for all x {\displaystyle x} in X {\displaystyle X} , then the function is said to be bounded (from) above by A {\displaystyle A} . If f ( x ) ≥ B {\displaystyle f(x)\geq B} for all x {\displaystyle x} in X {\displaystyle X} , then the function is said to be bounded (from) below by B {\displaystyle B} . A real-valued function is bounded if and only if it is bounded from above and below.[additional citation(s) needed]

An important special case is a bounded sequence, where X {\displaystyle X} is taken to be the set N {\displaystyle \mathbb {N} } of natural numbers. Thus a sequence f = ( a 0 , a 1 , a 2 , … ) {\displaystyle f=(a_{0},a_{1},a_{2},\ldots )} is bounded if there exists a real number M {\displaystyle M} such that

| a n | ≤ M {\displaystyle |a_{n}|\leq M}

for every natural number n {\displaystyle n} . The set of all bounded sequences forms the sequence space l ∞ {\displaystyle l^{\infty }} .

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

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

A bounded operator T : X → Y {\displaystyle T:X\rightarrow Y} is not a bounded function in the sense of this page's definition (unless T = 0 {\displaystyle T=0} ), but has the weaker property of preserving boundedness; bounded sets M ⊆ X {\displaystyle M\subseteq X} are mapped to bounded sets T ( M ) ⊆ Y {\displaystyle T(M)\subseteq Y} . This definition can be extended to any function f : X → Y {\displaystyle f:X\rightarrow Y} if X {\displaystyle X} and Y {\displaystyle 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 {\displaystyle \sin :\mathbb {R} \rightarrow \mathbb {R} } is bounded since | sin ⁡ ( x ) | ≤ 1 {\displaystyle |\sin(x)|\leq 1} for all x ∈ R {\displaystyle x\in \mathbb {R} } .34
  • The function f ( x ) = ( x 2 − 1 ) − 1 {\displaystyle f(x)=(x^{2}-1)^{-1}} , defined for all real x {\displaystyle x} except for −1 and 1, is unbounded. As x {\displaystyle 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 , ∞ ) {\displaystyle [2,\infty )} or ( − ∞ , − 2 ] {\displaystyle (-\infty ,-2]} .
  • The function f ( x ) = ( x 2 + 1 ) − 1 {\textstyle f(x)=(x^{2}+1)^{-1}} , defined for all real x {\displaystyle x} , is bounded, since | f ( x ) | ≤ 1 {\textstyle |f(x)|\leq 1} for all x {\displaystyle x} .
  • The inverse trigonometric function arctangent defined as: y = arctan ⁡ ( x ) {\displaystyle y=\arctan(x)} or x = tan ⁡ ( y ) {\displaystyle x=\tan(y)} is increasing for all real numbers x {\displaystyle x} and bounded with − π 2 < y < π 2 {\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}} radians5
  • By the boundedness theorem, every continuous function on a closed interval, such as f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\rightarrow \mathbb {R} } , is bounded.6 More generally, any continuous function from a compact space into a metric space is bounded.
  • All complex-valued functions f : C → C {\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} } which are entire are either unbounded or constant as a consequence of Liouville's theorem.7 In particular, the complex sin : C → C {\displaystyle \sin :\mathbb {C} \rightarrow \mathbb {C} } must be unbounded since it is entire.
  • The function f {\displaystyle f} which takes the value 0 for x {\displaystyle x} rational number and 1 for x {\displaystyle 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 ] {\displaystyle [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 : R 2 → R {\displaystyle g:\mathbb {R} ^{2}\to \mathbb {R} } and h : ( 0 , 1 ) 2 → R {\displaystyle h:(0,1)^{2}\to \mathbb {R} } defined by g ( x , y ) := x + y {\displaystyle g(x,y):=x+y} and h ( x , y ) := 1 x + y {\displaystyle h(x,y):={\frac {1}{x+y}}} are both continuous, but neither is bounded.8 (However, a continuous function must be bounded if its domain is both closed and bounded.9)

See also

References

  1. Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. 978-0-412-62150-5 ↩

  2. Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. 978-0-412-62150-5 ↩

  3. Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. 978-0-412-62150-5 ↩

  4. "The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021. https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf ↩

  5. 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. 978-1-4398-0640-1 ↩

  6. Weisstein, Eric W. "Extreme Value Theorem". mathworld.wolfram.com. Retrieved 2021-09-01. https://mathworld.wolfram.com/ExtremeValueTheorem.html ↩

  7. "Liouville theorems - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-09-01. https://encyclopediaofmath.org/wiki/Liouville_theorems ↩

  8. 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. 978-1-4419-1621-1 ↩

  9. 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. 978-1-4419-1621-1 ↩