Bounded Functions
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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |f(x)|\le M}
for all x in X.[1] A function that is not bounded is said to be unbounded.Template:Citation needed
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.[1]Template:Additional citation needed
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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|\le M}
for every natural number n. The set of all bounded sequences forms the sequence space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l^\infty} .Template:Citation needed
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.Template:Citation needed
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\sin (x)| \le 1} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbf{R}} .
- The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f(x)= (x^2+1)^{-1}} , defined for all real x, is bounded.
- The inverse trigonometric function arctangent defined as: y = arctan(x) or x = tan(y) is increasing for all real numbers x and bounded with −Template:Sfrac < y < Template:Sfrac 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g:\mathbb{R}^2\to\mathbb{R}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h: (0, 1)^2\to\mathbb{R}} defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x, y) := x + y} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x, y) := \frac{1}{x+y}} 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:
- Bounded function, Wikipedia under a CC BY-SA license
