well-behaved
well-behaved (phr)
English Definitions:
well-behaved, well behaved (adj)
(usually of children) someone who behaves in a manner that the speaker believes is correct
"a well-behaved child"
well-behaved (Adjective)
Having good manners and acting properly
well-behaved (Adjective)
(of a function) Having a finite derivative (of all orders) at all points, and having no discontinuities
Well-behaved
Mathematicians 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, 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 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. ⁕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.
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.
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"well-behaved." Kamus.net. STANDS4 LLC, 2024. Web. 24 Apr. 2024. <https://www.kamus.net/english/well-behaved>.
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