Bounded function
Template:More citations needed
In mathematics, a function <math>f</math> defined on some set <math>X</math> with real or complex values is called bounded if the set of its values (its image) is 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">Template:Cite book</ref> A function that is not bounded is said to be unbounded.Template:Citation needed
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" />Template:Additional citation needed
An important special case is a bounded sequence, where <math>X</math> is taken to be the set <math>\mathbb N</math> of natural numbers. 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>.Template:Citation needed
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>.Template:Citation needed
Related notionsEdit
Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.
A bounded operator <math>T: X \rightarrow Y</math> is not a bounded function in the sense of this page's definition (unless <math>T=0</math>), but has the weaker property of preserving boundedness; bounded sets <math>M \subseteq X</math> are mapped to bounded sets <math>T(M) \subseteq Y</math>. This definition can be extended to any function <math>f: X \rightarrow Y</math> if <math>X</math> and <math>Y</math> allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.Template:Citation needed
ExamplesEdit
- The sine function <math>\sin: \mathbb R \rightarrow \mathbb R</math> is bounded since <math>|\sin (x)| \le 1</math> for all <math>x \in \mathbb{R}</math>.<ref name=":0" /><ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref>
- The function <math>f(x)=(x^2-1)^{-1}</math>, defined for all real <math>x</math> except for −1 and 1, is unbounded. As <math>x</math> 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, <math>[2, \infty)</math> or <math>(-\infty, -2]</math>.Template:Citation needed
- The function <math display="inline">f(x)= (x^2+1)^{-1}</math>, defined for all real <math>x</math>, is bounded, since <math display="inline">|f(x)| \le 1</math> for all <math>x</math>.Template:Citation needed
- The inverse trigonometric function arctangent defined as: <math>y= \arctan (x)</math> or <math>x = \tan (y)</math> is increasing for all real numbers <math>x</math> and bounded with <math>-\frac{\pi}{2} < y < \frac{\pi}{2}</math> radians<ref>Template:Cite book</ref>
- By the boundedness theorem, every continuous function on a closed interval, such as <math>f: [0, 1] \rightarrow \mathbb R</math>, is bounded.<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> More generally, any continuous function from a compact space into a metric space is bounded.Template:Citation needed
- All complex-valued functions <math>f: \mathbb C \rightarrow \mathbb C</math> which are entire are either unbounded or constant as a consequence of Liouville's theorem.<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref> In particular, the complex <math>\sin: \mathbb C \rightarrow \mathbb C</math> must be unbounded since it is entire.Template:Citation needed
- The function <math>f</math> which takes the value 0 for <math>x</math> rational number and 1 for <math>x</math> 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 <math>[0, 1]</math> is much larger than the set of continuous functions on that interval.Template:Citation needed 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.<ref name=":1">Template:Cite book</ref> (However, a continuous function must be bounded if its domain is both closed and bounded.<ref name=":1" />)