Bounded Functions - Revision history

Lila: /* Boundedness Theorem */

2021-11-17T21:56:45Z

Boundedness Theorem

Lila: /* Boundedness Theorem */

2021-11-17T21:56:01Z

Boundedness Theorem

Lila: /* Example 1 */

2021-11-17T21:53:46Z

Example 1

Lila: /* Boundedness Theorem */

2021-11-17T21:53:13Z

Boundedness Theorem

Lila: /* Examples */

2021-11-17T21:51:24Z

Examples

Lila at 21:45, 17 November 2021

2021-11-17T21:45:46Z

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2021-11-17T21:44:41Z

Created page with "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 f..."

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[[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.]]
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
:<math>|f(x)|\le M</math>
for all ''x'' in ''X''.<ref name=":0">{{Cite book|last=Jeffrey|first=Alan|url=https://books.google.com/books?id=jMUbUCUOaeQC&newbks=0&printsec=frontcover&pg=PA66&dq=%22Bounded+function%22&hl=en|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 ''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.<ref name=":0" />{{Additional citation needed|date=September 2021}}

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

:<math>|a_n|\le M</math>
for every natural number ''n''. 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 ''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|date= September 2021}}

== Related notions ==
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.

==Examples==
* The sine function sin : '''R''' → '''R''' is bounded since <math>|\sin (x)| \le 1</math> for all <math>x \in \mathbf{R}</math>.

* The function <math>f(x)=(x^2-1)^{-1}</math>, 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].

* The function <math display="inline">f(x)= (x^2+1)^{-1}</math>, defined for all real ''x'', ''is'' bounded.

* The inverse trigonometric function arctangent defined as: ''y'' = {{math|arctan(''x'')}} or ''x'' = {{math|tan(''y'')}} is increasing for all real numbers ''x'' and bounded with −{{sfrac|{{pi}}|2}} < ''y'' < {{sfrac|{{pi}}|2}} radians.
* By the boundedness theorem, every continuous function on a closed interval, such as ''f'' : [0, 1] → '''R''', is bounded. More generally, any continuous function from a compact space into a metric space is bounded.

*All complex-valued functions ''f'' : '''C''' → '''C''' which are entire are either unbounded or constant as a consequence of Liouville's theorem. In particular, the complex sin : '''C''' → '''C''' must be unbounded since it is entire.

* 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. Moreover, continuous functions need not be bounded; for example, the functions <math>g:\mathbb{R}^2\to\mathbb{R}</math> and <math>h: (0, 1)^2\to\mathbb{R}</math> defined by <math>g(x, y) := x + y</math> and <math>h(x, y) := \frac{1}{x+y}</math> are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Bounded_function Bounded function, Wikipedia] under a CC BY-SA license

@@ Line 38: / Line 38: @@
 <td><strong>Theorem 1 (Boundedness):</strong> If <span class="math-inline"><math>I = [a, b]</math></span> is a closed and bounded interval, and <span class="math-inline"><math>f : I \to \mathbb{R}</math></span> is a continuous function on <span class="math-inline"><math>I</math></span>, then <span class="math-inline"><math>f</math></span> is bounded on <span class="math-inline"><math>I</math></span>.
-[[File:Continuous function bounded on interval.png|thumb|Continuous function bounded on interval]]
+[[File:Continuous function bounded on interval.png|Continuous function bounded on interval]]
 </td>

@@ Line 37: / Line 37: @@
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <td><strong>Theorem 1 (Boundedness):</strong> If <span class="math-inline"><math>I = [a, b]</math></span> is a closed and bounded interval, and <span class="math-inline"><math>f : I \to \mathbb{R}</math></span> is a continuous function on <span class="math-inline"><math>I</math></span>, then <span class="math-inline"><math>f</math></span> is bounded on <span class="math-inline"><math>I</math></span>.
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/boundedness-theorem/Screen%20Shot%202014-11-27%20at%2011.29.09%20PM.png" alt="Screen%20Shot%202014-11-27%20at%2011.29.09%20PM.png" class="image" /></div>
 </td>
 </blockquote>

@@ Line 68: / Line 68: @@
 ===Example 1===
 <p><strong>Verify that the function <span class="math-inline"><math>f : [0, 2] \to \mathbb{R}</math></span> defined by <span class="math-inline"><math>f(x) = x^2</math></span> is bounded.</strong></p>
-<p>We first note that <span class="math-inline"><math>f</math></span> defined by <span class="math-inline"><math>f(x) = x^2</math></span> is continuous on all of <span class="math-inline"><math>\mathbb{R}</math></span> and so <span class="math-inline"><math>f</math></span> is also continuous on the interval <span class="math-inline"><math>[0, 2]</math></span>. Furthermore, <span class="math-inline"><math>I = [0, 2]</math></span> is a closed bounded interval. Since <span class="math-inline"><math>f</math></span> is an increasing function on the interval <span class="math-inline"><math>[0, 2]</math></span> as <span class="math-inline"><math>f' > 0</math></span> on <span class="math-inline"><math>[0, 2]</math></span> we conclude that <span class="math-inline"><math>\mid f(x) \mid ≤ 4</math></span> for all <span class="math-inline"><math>x \in [0, 2]</math></span>.</p>
+<p>We first note that <span class="math-inline"><math>f</math></span> defined by <span class="math-inline"><math>f(x) = x^2</math></span> is continuous on all of <span class="math-inline"><math>\mathbb{R}</math></span> and so <span class="math-inline"><math>f</math></span> is also continuous on the interval <span class="math-inline"><math>[0, 2]</math></span>. Furthermore, <span class="math-inline"><math>I = [0, 2]</math></span> is a closed bounded interval. Since <span class="math-inline"><math>f</math></span> is an increasing function on the interval <span class="math-inline"><math>[0, 2]</math></span> as <span class="math-inline"><math>f' > 0</math></span> on <span class="math-inline"><math>[0, 2]</math></span> we conclude that <span class="math-inline"><math>\mid f(x) \mid \leq 4</math></span> for all <span class="math-inline"><math>x \in [0, 2]</math></span>.</p>
 ===Example 2===
 <p><strong>Verify that the function <span class="math-inline"><math>f : [-1, 1] \to \mathbb{R}</math></span> defined by <span class="math-inline"><math>f(x) = \frac{1}{x}</math></span> does not satisfy the conditions of the boundedness theorem.</strong></p>

@@ Line 53: / Line 53: @@
 </ul>
 <ul>
-<li>But this is a contradiction. Notice that <span class="math-inline"><math>\mid f(x_{n_k}) \mid > n_{k} ≥ k</math></span> for all <span class="math-inline"><math>k \in \mathbb{N}</math></span>, and so our supposition that <span class="math-inline"><math>f</math></span> was not bounded on <span class="math-inline"><math>I</math></span> was false. Therefore <span class="math-inline"><math>f</math></span> is bounded on <span class="math-inline"><math>I</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+<li>But this is a contradiction. Notice that <span class="math-inline"><math>\mid f(x_{n_k}) \mid > n_{k} \geq k</math></span> for all <span class="math-inline"><math>k \in \mathbb{N}</math></span>, and so our supposition that <span class="math-inline"><math>f</math></span> was not bounded on <span class="math-inline"><math>I</math></span> was false. Therefore <span class="math-inline"><math>f</math></span> is bounded on <span class="math-inline"><math>I</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 </ul>
 <p>We should make special note that the conclusion to the boundedness theorem is guaranteed to hold provided that:</p>

@@ Line 31: / Line 31: @@
 * 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. Moreover, continuous functions need not be bounded; for example, the functions <math>g:\mathbb{R}^2\to\mathbb{R}</math> and <math>h: (0, 1)^2\to\mathbb{R}</math> defined by <math>g(x, y) := x + y</math> and <math>h(x, y) := \frac{1}{x+y}</math> are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.)
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Bounded_function Bounded function, Wikipedia] under a CC BY-SA license

@@ Line 2: / Line 2: @@
 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
 :<math>|f(x)|\le M</math>
-for all ''x'' in ''X''.<ref name=":0">{{Cite book|last=Jeffrey|first=Alan|url=https://books.google.com/books?id=jMUbUCUOaeQC&newbks=0&printsec=frontcover&pg=PA66&dq=%22Bounded+function%22&hl=en|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}}
+for all ''x'' in ''X''. A function that is ''not'' bounded is said to be '''unbounded'''.
-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.<ref name=":0" />{{Additional citation needed|date=September 2021}}
+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.
 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''<sub>0</sub>, ''a''<sub>1</sub>,  ''a''<sub>2</sub>, ...) is bounded if there exists a real number ''M'' such that
 :<math>|a_n|\le M</math>
-for every natural number ''n''. The set of all bounded sequences forms the sequence space <math>l^\infty</math>.{{Citation needed|date=September 2021}}
+for every natural number ''n''. The set of all bounded sequences forms the sequence space <math>l^\infty</math>.
-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|date= September 2021}}
+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''.
 == Related notions ==