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Well-behaved

[Mathematician] s (and those in related sciences) very frequently speak of whether a [mathematical] object — a [function] , a [set] , a space of one sort or another — is ** "well-behaved" ** or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste. To ensure that an object is "well-behaved" mathematicians introduce further axioms to narrow down the domain of study. This has the benefit of making analysis easier, but cuts down on the generality of any conclusions reached. Concepts like non-Euclidean geometry were once considered ill-behaved, but are now common objects of study.

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.

Despite the list below, in practice "well-behaved" is almost always used in an absolute sense.

Usually,

- In [calculus] :

-- [Analytic function] s are better-behaved than general [smooth function] s.

-- Smooth functions are better-behaved than general differentiable functions.

-- [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.

-- [Continuous function] s are better-behaved than [Riemann-integrable] functions on compact sets.

-- Riemann-integrable functions are better-behaved than [Lebesgue-integrable] functions.

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