Well-behaved

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Mathematicians (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". The term has no fixed formal definition, and is dependent on context, 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.

The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not. For example:

  • In Bézout's theorem, two polynomials are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if their polynomial greatest common divisor is a constant.

Unusually, the term could also be applied in a comparative sense: