Bounded function: Difference between revisions

Content deleted Content added

Inline

Revision as of 13:16, 14 January 2019

In mathematics, 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

|f(x)|\leq M

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.

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = (a₀, a₁, a₂, ...) is bounded if there exists a real number M such that

|a_{n}|\leq M

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

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

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.

Examples

The function sin : R → R is bounded.
The function $f(x)=(x^{2}-1)^{-1}$ 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 ${\textstyle f(x)=(x^{2}+1)^{-1}}$ defined for all real x is bounded.

Every continuous function 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 : R → 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's 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 bigger than the set of continuous functions on that interval.

@@ Line 5: / Line 5: @@
 [[for all]] ''x'' in ''X''. A function that is ''not'' bounded is said to be '''unbounded'''.
-Sometimes, if ''f''(''x'') ≤ ''A'' for all ''x'' in ''X'', then the function is said to be '''bounded above''' by ''A''. On the other hand, if ''f''(''x'') ≥ ''B'' for all ''x'' in ''X'', then the function is said to be '''bounded below''' by ''B''.
+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 number]]s. 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
-The concept should not be confused with that of a [[bounded operator]].
-An important special case is a '''bounded sequence''', where ''X'' is taken to be the set '''N''' of [[natural number]]s. 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, equipped with a [[vector space]] structure, forms a [[sequence space]].
+for every natural number ''n''. The set of all bounded sequences forms the [[sequence space]] <math>l^\infty</math>.
-This definition can be extended to functions taking values in a [[metric space]] ''Y''. Such a function ''f'' defined on some set ''X'' is called bounded if for some ''a'' in ''Y'' there exists a real number ''M'' such that its distance function ''d'' ("distance") is less than ''M'', i.e.
+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 ==
-:<math>d(f(x), a)\le M</math>
+Weaker than boundedness is [[local boundedness]]. A family of bounded functions may be [[Uniform boundedness|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.
-for all ''x'' in ''X''.
-If this is the case, there is also such an ''M'' for each other ''a'', by the triangle inequality.
 ==Examples==
-* The function ''f'' : '''R''' → '''R''' defined by ''f''(''x'') = sin(''x'') is bounded. The [[sine]] function is no longer bounded if it is defined over the set of all complex numbers.
+* The function sin : '''R''' → '''R''' is bounded.
+* 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>f(x)=\frac{1}{x^2-1}</math>
+* The function <math display="inline">f(x)= (x^2+1)^{-1}</math>defined for all real ''x'' ''is'' bounded.
-: defined for all real ''x'' except for −1 and 1 is unbounded. As ''x'' gets closer to −1 or to 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
+* Every [[continuous function]] ''f'' : [0, 1] → '''R''' is bounded. More generally, any continuous function from a [[compact space]] into a metric space is bounded.
-:: <math>f(x)=\frac{1}{x^2+1}</math>
+*All complex-valued functions ''f'' : '''R''' → '''C''' which are [[Entire function|entire]] are either unbounded or constant as a consequence of [[Liouville's theorem (complex analysis)|Liouville's theorem]]. In particular, the complex sin : '''C''' → '''C''' must be unbounded since it's entire.
-: defined for all real ''x'' ''is'' bounded.
-* Every [[continuous function]] ''f'' : [0, 1] → '''R''' is bounded.   This is really a special case of a more general fact:  Every continuous function from a [[compact space]] into a metric space is bounded.
-** Note that for example <math>f(x) \neq \frac{1}{x}</math>, because <math>\lim_{h\rightarrow 0}f(h) = \infty \notin \mathbb{R}</math>
 * The function ''f'' which takes the value 0 for ''x'' [[rational number]] and 1 for ''x'' [[irrational number]] (cf. [[Nowhere continuous function#Dirichlet function|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 bigger than the set of [[continuous function]]s on that interval.
@@ Line 37: / Line 32: @@
 * [[Bounded set]]
 * [[Support (mathematics)#Compact support|Compact support]]
+*[[Local boundedness]]
 * [[Uniform boundedness]]

Revision as of 13:16, 14 January 2019

Related Notions

Examples

See also