Bounded function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 defined on some <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> values is called bounded if the set of its values is <a href="/facts/Bounded_set/yCldjtoy">bounded</a>. In other words, <a href="/facts/There_exists/hHveaBYo">there exists</a> a real number 
 
 
 
 M
 
 
 {\displaystyle M}
 
 such that

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

<a href="/facts/For_all/agNmvJKN">for all</a> 
 
 
 
 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 <a href="/facts/Natural_number/u65og8JE">natural numbers</a>. Thus a <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 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 <a href="/facts/Sequence_space/2HkcM12K">sequence space</a> 
 
 
 
 
 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 <a href="/facts/Bounded_set/yCldjtoy">bounded set</a> in 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
.

Bounded function open-in-new

Bounded function