Well-behaved

Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- 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 being 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 of the assumptions needed for the successful application of whatever analysis is being discussed.

The opposite case is usually labelled pathological.

Generally,

• Continuous functions are better-behaved than discontinuous ones.
• Differentiable functions are better-behaved than general continuous functions.
• 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.
• No one can agree on whether the axiom of choice is well-behaved or pathological.

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