Editing Bounded function (section)

{{Short description|A mathematical function the set of whose values is bounded}}
{{More citations needed|date=September 2021}}[[Image:Bounded and unbounded functions.svg|right|thumb|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.]]
In [[mathematics]], a [[function (mathematics)|function]] <math>f</math> defined on some [[Set (mathematics)|set]] <math>X</math> with [[real number|real]] or [[complex number|complex]] values is called '''bounded''' if the set of its values (its [[Image (mathematics)|image]]) is [[bounded set|bounded]]. In other words, [[there exists]] a real number <math>M</math> such that 
:<math>|f(x)|\le M</math>
[[for all]] <math>x</math> in <math>X</math>.<ref name=":0">{{Cite book|last=Jeffrey|first=Alan|url=https://books.google.com/books?id=jMUbUCUOaeQC&dq=%22Bounded+function%22&pg=PA66|title=Mathematics for Engineers and Scientists, 5th Edition|date=1996-06-13|publisher=CRC Press|isbn=978-0-412-62150-5|language=en}}</ref> A function that is ''not'' bounded is said to be '''unbounded'''.{{Citation needed|date=September 2021}}

If <math>f</math> is real-valued and <math>f(x) \leq A</math> for all <math>x</math> in <math>X</math>, then the function is said to be '''bounded (from) above''' by <math>A</math>. If <math>f(x) \geq B</math> for all <math>x</math> in <math>X</math>, then the function is said to be '''bounded (from) below''' by <math>B</math>. A real-valued function is bounded if and only if it is bounded from above and below.<ref name=":0" />{{Additional citation needed|date=September 2021}}

An important special case is a '''bounded sequence''', where ''<math>X</math>'' is taken to be the set <math>\mathbb N</math> of [[natural number]]s. Thus a [[sequence]] <math>f = (a_0, a_1, a_2, \ldots)</math> is bounded if there exists a real number <math>M</math> such that

:<math>|a_n|\le M</math>
for every natural number <math>n</math>. The set of all bounded sequences forms the [[sequence space]] <math>l^\infty</math>.{{Citation needed|date=September 2021}}

The definition of boundedness can be generalized to functions <math>f: X \rightarrow Y</math> taking values in a more general space <math>Y</math> by requiring that the image <math>f(X)</math> is a [[bounded set]] in <math>Y</math>.{{Citation needed|date= September 2021}}