Editing Pathological (mathematics) (section)

== Well-behaved ==

[[Mathematician]]s (and those in related sciences) very frequently speak of whether a [[mathematics|mathematical]] object&mdash;a [[Function (mathematics)|function]], a [[Set (mathematics)|set]], a [[Space (mathematics)|space]] of one sort or another&mdash;is '''"well-behaved"'''. While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be 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 produces a [[loss of generality]] of any conclusions reached.

In both pure and applied mathematics (e.g., [[Optimization (mathematics)|optimization]], [[numerical integration]], [[mathematical physics]]), ''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]] or [[measure (mathematics)|measure]]) 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&mdash;either something is well-behaved or it is not. For example:

*In [[algorithmic inference]], a [[well-behaved statistic]] is monotonic, well-defined, and [[Sufficient statistic|sufficient]].
*In [[Bézout's theorem]], two [[polynomial]]s 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.
*A [[meromorphic function]] is a ratio of two well-behaved functions, in the sense of those two functions being [[Holomorphic function|holomorphic]].
*The [[Karush–Kuhn–Tucker conditions]] are first-order necessary conditions for a solution in a well-behaved [[nonlinear programming]] problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied.
*In [[probability]], events contained in the [[probability space]]'s corresponding [[sigma-algebra]] are well-behaved, as are [[measurable]] functions.

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

*In [[calculus]]:
**[[Analytic function]]s are better-behaved than general [[smooth function]]s.
**Smooth functions are better-behaved than general differentiable functions.
**Continuous [[Differentiable function|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 integration|Riemann-integrable]] functions on compact sets.
**Riemann-integrable functions are better-behaved than [[Lebesgue integration|Lebesgue-integrable]] functions.
**Lebesgue-integrable functions are better-behaved than general functions.
*In [[topology]]:
**[[Continuous function (topology)|Continuous]] functions are better-behaved than discontinuous ones.
**[[Euclidean space]] is better-behaved than [[non-Euclidean geometry]].
**Attractive [[fixed point (mathematics)|fixed points]] are better-behaved than repulsive fixed points.
**[[Hausdorff topology|Hausdorff topologies]] are better-behaved than those in arbitrary [[general topology]].
**[[Borel set]]s are better-behaved than arbitrary [[set (mathematics)|sets]] of [[real number]]s.
**Spaces with [[integer]] dimension are better-behaved than spaces with [[fractal dimension]].
*In [[abstract algebra]]:
**[[Group (mathematics)|Groups]] are better-behaved than [[Magma (algebra)|magmas]] and [[semigroup]]s.
**[[Abelian group]]s are better-behaved than non-Abelian groups.
**[[Finitely-generated abelian group|Finitely-generated Abelian group]]s are better-behaved than non-finitely-generated Abelian groups.
**[[wiktionary:finite|Finite]]-[[dimension (vector space)|dimensional]] [[vector space]]s are better-behaved than [[Infinity|infinite]]-dimensional ones.
**[[Field (mathematics)|Fields]] are better-behaved than [[skew field]]s or general [[ring (mathematics)|rings]].
**Separable [[field extension]]s are better-behaved than non-separable ones.
**[[Normed division algebra]]s are better-behaved than general composition algebras.