Continuity property

In mathematics, the continuity property may be presented as follows.

:Suppose that "f" : ["a", "b"] → R is a continuous function. Then the image "f"( ["a", "b"] ) is a closed bounded interval.

The theorem is the combination of the intermediate value theorem and the extreme value theorem, comprising the three assertions:
# The image "f"( ["a", "b"] ) is an interval.
# This image is bounded.
# This image interval is closed, so "f" attains both its bounds.

The first assertion is broadly the intermediate value theorem; the latter two are equivalent to the extreme value theorem.

Proof of assertion 1

"See:" Intermediate value theorem#Proof

Proof of assertion 2

"See:" Extreme value theorem#Proof of the boundedness theorem

Proof of assertion 3

"See:" Extreme value theorem#Proof of the extreme value theorem

Caveats

It is important to note that this theorem only applies to continuous real functions. It does not apply to functions with function domain the rational numbers. As the rationals do not satisfy the least upper bound axiom, they are not complete.

To illustrate, this consider

f: [0,2] cap mathbb{Q} o mathbb{R}

x mapsto e^{( - (x - sqrt{2})^2 )}

f would obtain its maximum value at sqrt{2}but this is not in the set.

If f is not continuous consider as a counterexample

f: [0,1] o mathbb{R}

x mapsto egin{cases}frac{1}{x} & frac{1}{x} in mathbb{Z} \0 & mbox{otherwise}end{cases}

This is unbounded, but [0,1] is bounded.

Further, one should carefully note that the set must be closed, otherwise the maximum and minimum values might not be obtained.

References

* Binmore, K. G. "Mathematical Analysis: A Straightforward Approach", Cambridge University Press, (1982). ISBN 0521288827.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Property law — is the area of law that governs the various forms of ownership in real property (land as distinct from personal or movable possessions) and in personal property, within the common law legal system. In the civil law system, there is a division… …   Wikipedia

  • property — and property rights are central to capitalist societies. Perhaps because they are largely taken for granted in this context they have received relatively little attention from sociologists. By comparison, political philosophers and economists… …   Dictionary of sociology

  • Continuity (fiction) — In fiction, continuity (also called time scheme) is consistency of the characteristics of persons, plot, objects, places and events seen by the reader or viewer over some period of time. It is of relevance to several media. Continuity is… …   Wikipedia

  • continuity — /kon tn ooh i tee, tn yooh /, n., pl. continuities. 1. the state or quality of being continuous. 2. a continuous or connected whole. 3. a motion picture scenario giving the complete action, scenes, etc., in detail and in the order in which they… …   Universalium

  • Property type (National Register of Historic Places) — The U.S. National Register of Historic Places (NRHP) classifies its listings by various types of properties. Listed properties generally fall into one of five categories, though there are special considerations for other types of properties which …   Wikipedia

  • Property master — The property master is an artistic and organizational employee in a film, television or theatrical production who is responsible for purchasing, acquiring and/or manufacturing any props needed for a production. The property master also works with …   Wikipedia

  • continuity — noun (plural ties) Date: 15th century 1. a. uninterrupted connection, succession, or union b. uninterrupted duration or continuation especially without essential change 2 …   New Collegiate Dictionary

  • continuity — noun /ˌkɒntɪˈnjuːəti,ˌkɑntɪˈn(j)uːəti/ a) Lack of interruption or disconnection; the quality of being continuous in space or time. Considerable continuity of attention is needed to read German philosophy. b) A characteristic property of a… …   Wiktionary

  • continuity — con•ti•nu•i•ty [[t]ˌkɒn tnˈu ɪ ti, tnˈyu[/t]] n. pl. ties 1) the state or quality of being continuous 2) a continuous or connected whole 3) sbz a motion picture scenario with all details of the action, dialogue, effects, etc., in order 4) sbz (on …   From formal English to slang

  • Modulus of continuity — In mathematical analysis, a modulus of continuity is a function used to measure quantitatively the uniform continuity of functions. So, a function admits ω as a modulus of continuity if and only if for all x and y in the domain of f. Since moduli …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”