Bounded function

<h2 id="related-notions">Related notions</h2>
Weaker than boundedness is <a href="/facts/Local_boundedness/stbG6j9j">local boundedness</a>. A family of bounded functions may be <a href="/facts/Uniform_boundedness/nyxwk4Ix">uniformly bounded</a>. 
A <a href="/facts/Bounded_operator/LYmDTIqR">bounded operator</a> 
 
 
 
 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.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Sine/gQ6LXK8z">sine</a> 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} }
 
.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>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]}
 
.</li>
<li>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}
 
.</li>
<li>The <a href="/facts/Inverse_trigonometric_function/bVwU9FPG">inverse trigonometric function</a> arctangent defined as: 
 
 
 
 y
 =
 arctan
 ⁡
 (
 x
 )
 
 
 {\displaystyle y=\arctan(x)}
 
 or 
 
 
 
 x
 =
 tan
 ⁡
 (
 y
 )
 
 
 {\displaystyle x=\tan(y)}
 
 is <a href="/facts/Monotonic_function/MjQK0YL2">increasing</a> for all real numbers 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and bounded with 
 
 
 
 −
 
 
 π
 2
 
 
 <
 y
 <
 
 
 π
 2
 
 
 
 
 {\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}
 
 <a href="/facts/Radian/Srd38Sxa">radians</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li>
<li>By the <a href="/facts/Boundedness_theorem/eSV6P7eh">boundedness theorem</a>, every <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> on a closed interval, such as 
 
 
 
 f
 :
 [
 0
 ,
 1
 ]
 →
 
 R
 
 
 
 {\displaystyle f:[0,1]\rightarrow \mathbb {R} }
 
, is bounded.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> More generally, any continuous function from a <a href="/facts/Compact_space/d0cgXJH7">compact space</a> into a metric space is bounded.</li>
<li>All complex-valued functions 
 
 
 
 f
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }
 
 which are <a href="/facts/Entire_function/XNluKzu7">entire</a> are either unbounded or constant as a consequence of <a href="/facts/Liouville%27s_theorem_(complex_analysis)/I97twMBQ">Liouville's theorem</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In particular, the complex 
 
 
 
 sin
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle \sin :\mathbb {C} \rightarrow \mathbb {C} }
 
 must be unbounded since it is entire.</li>
<li>The function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 which takes the value 0 for 
 
 
 
 x
 
 
 {\displaystyle x}
 
 <a href="/facts/Rational_number/VJQmyY05">rational number</a> and 1 for 
 
 
 
 x
 
 
 {\displaystyle x}
 
 <a href="/facts/Irrational_number/Psv19TET">irrational number</a> (cf. <a href="/facts/Nowhere_continuous_function/XbxlDzx1">Dirichlet function</a>) is bounded. Thus, a function <a href="/facts/Pathological_(mathematics)/KmKHjJ1H">does not need to be "nice"</a> 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 <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> 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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> (However, a continuous function must be bounded if its domain is both closed and bounded.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>)</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bounded_set/yCldjtoy">Bounded set</a></li>
<li><a href="/facts/Support_(mathematics)/BTuMGbsb">Compact support</a></li>
<li><a href="/facts/Local_boundedness/stbG6j9j">Local boundedness</a></li>
<li><a href="/facts/Uniform_boundedness/nyxwk4Ix">Uniform boundedness</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. <a href="978-0-412-62150-5" target="_blank">978-0-412-62150-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. <a href="978-0-412-62150-5" target="_blank">978-0-412-62150-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Jeffrey, Alan (1996-06-13). Mathematics for Engineers and Scientists, 5th Edition. CRC Press. ISBN 978-0-412-62150-5. <a href="978-0-412-62150-5" target="_blank">978-0-412-62150-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"The Sine and Cosine Functions" (PDF). math.dartmouth.edu. Archived (PDF) from the original on 2 February 2013. Retrieved 1 September 2021. <a href="https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf" target="_blank">https://math.dartmouth.edu/opencalc2/cole/lecture10.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-1-4398-0640-1" target="_blank">978-1-4398-0640-1</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Weisstein, Eric W. "Extreme Value Theorem". mathworld.wolfram.com. Retrieved 2021-09-01. <a href="https://mathworld.wolfram.com/ExtremeValueTheorem.html" target="_blank">https://mathworld.wolfram.com/ExtremeValueTheorem.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"Liouville theorems - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-09-01. <a href="https://encyclopediaofmath.org/wiki/Liouville_theorems" target="_blank">https://encyclopediaofmath.org/wiki/Liouville_theorems</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-1-4419-1621-1" target="_blank">978-1-4419-1621-1</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-1-4419-1621-1" target="_blank">978-1-4419-1621-1</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Bounded function open-in-new

Bounded function