Well-behaved

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 Function s are better-behaved than general differentiable functions.

Analytic Function s are better-behaved than general smooth functions

Euclidean Space is better-behaved than Non-Euclidean Geometry .

Attractive Fixed Point s are better-behaved than repulsive fixed points.

Field s are better-behaved than Skew Field s.

Hausdorff Topologies are better-behaved than those in arbitrary General Topology .

Separable Field Extension s are better-behaved than Non-separable Ones .

Borel Set s are better-behaved than arbitrary Set s of Real Number s.

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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Well-behaved