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# Well-behaved

Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object &mdash; a number, a function, a set, a space of one sort or another &mdash; is "well-behaved" or not. While the term has no fixed formal definition, it can have fairly precise meaning within a given context.

In pure mathematics, "well-behaved" objects are those that can be proved or analyzed by elegant means to have elegant properties.

In both pure and applied mathematics (optimization, numerical integration, or mathematical physics, for example), "well-behaved" also means not violating any assumptions needed to successfully apply whatever analysis is being discussed.

The opposite case is usually labeled pathological. It is not unusual to have situations in which most cases (in terms of cardinality) are pathological, but the pathological cases will not arise in practice unless constructed deliberately. (Of course, in these matters of taste one person's "well-behaved" vs. "pathological" dichotomy is usually some other person's division into "trivial" vs. "interesting".)

Generally,

*Lebesgue-integrable functions are better-behaved than general functions in calculus.
*Riemann-integrable functions are better-behaved than Lebesgue-integrable functions in calculus.
*Continuous functions are better-behaved than Riemann-integrable functions on compact sets in calculus.
*Continuous functions are better-behaved than discontinuous ones in topology.
*Differentiable functions are better-behaved than general continuous functions. The larger the number of times the function can be differentiated, the more well-behaved it is.
*Smooth functions are better-behaved than general differentiable functions.
*Analytic functions are better-behaved than general smooth functions
*Euclidean space is better-behaved than non-Euclidean geometry.
*Attractive fixed points are better-behaved than repulsive fixed points.
*Fields are better-behaved than skew fields.
*Hausdorff topologies are better-behaved than those in arbitrary general topology.
*Separable field extensions are better-behaved than non-separable ones.
*Borel sets are better-behaved than arbitrary sets of real numbers.
*Spaces with integer dimension are better-behaved than spaces with fractal dimension.
*Spaces with dimension are better-behaved than spaces with infinite dimension in linear algebra.

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Synonyms:

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• well-behaved — well be haved adj behaving in a calm polite way, and not being rude or violent ▪ a well behaved child ▪ a very well behaved dog ▪ The crowd was noisy but well behaved …   Dictionary of contemporary English

• well-behaved — 1590s, from WELL (Cf. well) (adv.) + pp. of BEHAVE (Cf. behave) …   Etymology dictionary

• well-behaved — well be|haved [ ,wel bı heıvd ] adjective a well behaved child or animal behaves in a way that is polite or gentle and does not upset people …   Usage of the words and phrases in modern English

• well-behaved — [wel΄bē hāvd′] adj. behaving well; conducting oneself properly; displaying good manners …   English World dictionary

• well-behaved — index civil (polite), malleable, obedient Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

• well-behaved — well′ behaved′ adj …   From formal English to slang

• well behaved — adjective (usually of children) someone who behaves in a manner that the speaker believes is correct a well behaved child • Syn: ↑well behaved • Similar to: ↑good * * * ▶ adjective ORDERLY, obedient, disciplined, peaceable, docile, controlled,… …   Useful english dictionary

• well-behaved — also well behaved ADJ GRADED If you describe someone, especially a child, as well behaved, you mean that they behave in a way that adults generally like and think is correct. ...well behaved little boys... The troops have been remarkably well… …   English dictionary

• well-behaved — adjective behaving in a polite or socially acceptable way: a well behaved child | The crowd was noisy but well behaved …   Longman dictionary of contemporary English

• well-behaved — adjective (usually of children) someone who behaves in a manner that the speaker believes is correct a well behaved child • Syn: ↑well behaved • Similar to: ↑good …   Useful english dictionary