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In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. The classical case is probably that of some everywhere continuous functions that are in fact nowhere differentiable, such as the Weierstrass function. In that case, the Baire category theorem was later used to show, quite to the contrary, that such behaviour was typical and even generic.
Often the usefulness of a theorem is justified by saying examples which don't meet the assumptions (counterexamples) are pathological. A famous case is the Alexander horned sphere, a counterexample showing that embedding topologically a sphere S2 in R3 may fail to separate the space cleanly, unless an extra condition of tameness is used to suppress possible wild behaviour.
One can therefore say that (particularly in mathematical analysis) those searching for the 'pathological' are like experimentalists, interested in knocking down potential theorems proposed (by 'theorists'); this should all take place within mathematics. What is created especially can have some undesirable, unusual, or other properties that make it difficult to contain or explain within a theory. But that point of view is probably biased by preconceptions.
Pathological examples can show the importance of the assumptions in a theorem. For example, in statistics, the Cauchy distribution does not satisfy the Central limit theorem, even though its symmetric bell-shape appears similar to many distributions which do; it fails the requirement to have a mean and standard deviation which exist and are finite.
The best-known paradoxes such as the Banach-Tarski paradox and Hausdorff paradox are based on the existence of non-measurable sets. Mathematicians, unless they take the minority position of denying the axiom of choice, are in general resigned to living with such sets.
Other examples include the Peano space filling curve which maps the unit interval [0,1] continuously onto [0,1] × [0,1], and the Cantor set which is a subset of the interval [0,1] and has the pathological property that it is uncountable yet its measure is zero.
See also: well-behaved.
This article incorporates material from pathological (http://planetmath.org/?op=getobj&from=objects&id=6310) on PlanetMath, which is licensed under the GFDL.
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