Well-Behaved Function
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A Well-Behaved Function is a function in a relationship with some other function that is more difficult to analyze.
- Example(s):
- a Continuous Function in relation to a Discontinuous Function.
- a Differentiable Function in relation to a Non-Differentiable Function.
- See: Well-Behaved, Function, Pathological Function.
References
2012
- (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Well-behaved
- 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 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 applied in an absolute sense.
- 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 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 calculus:
- Analytic functions are better-behaved than general smooth functions.
- Smooth functions are better-behaved than general differentiable functions.
- Continuous 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 functions are better-behaved than Riemann-integrable functions on compact sets.
- Riemann-integrable functions are better-behaved than Lebesgue-integrable functions.
- Lebesgue-integrable functions are better-behaved than general functions.
- In topology, continuous functions are better-behaved than discontinuous ones.
- Euclidean space is better-behaved than non-Euclidean geometry.
- Attractive fixed points are better-behaved than repulsive fixed points.
- Hausdorff topologies are better-behaved than those in arbitrary general topology.
- Borel sets are better-behaved than arbitrary sets of real numbers.
- Spaces with integer dimension are better-behaved than spaces with fractal dimension.
- Finite-dimensional vector spaces are better-behaved than infinite-dimensional ones.
- In abstract algebra:
- Fields are better-behaved than skew fields or general rings.
- Separable field extensions are better-behaved than non-separable ones.
- Normed division algebras are better-behaved than general composition algebras.