Editing Pathological (mathematics) (section)

=={{anchor|Pathological example}}Pathological examples==
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{{original research|date=August 2019}}
Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviors often prompt new investigation and research, which leads to new theory and more general results. Some important historical examples of this are:

*{{anchor|In voting}}[[Instant-runoff voting|Ranked-choice voting]] is commonly described as a pathological [[social choice function]], because of its tendency to eliminate candidates for [[Monotonicity criterion|winning too many votes]].<ref name=":0">{{Cite journal |last1=Doron |first1=Gideon |last2=Kronick |first2=Richard |date=1977 |title=Single Transferrable Vote: An Example of a Perverse Social Choice Function |url=https://www.jstor.org/stable/2110496 |journal=American Journal of Political Science |volume=21 |issue=2 |pages=303–311 |doi=10.2307/2110496 |jstor=2110496 |issn=0092-5853}}</ref>
*The discovery of [[irrational number]]s by the school of [[Pythagoras]] in ancient Greece; for example, the length of the diagonal of a [[unit square]], that is <math>\sqrt{2}</math>.
*The discovery of [[complex number]]s in the 16th century in order to find the roots of [[Cubic function|cubic]] and [[Quartic function|quartic]] [[polynomial function]]s. 
*Some [[number field]]s have [[ring of integers|rings of integers]] that do not form a [[unique factorization domain]], for example the [[Field extension|extended field]] <math>\mathbb{Q}(\sqrt{-5})</math>.
*The discovery of [[fractal]]s and other "rough" geometric objects (see [[Hausdorff dimension]]).
*[[Weierstrass function]], a [[real number|real]]-valued function on the [[real line]], that is [[continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere.<ref name=":1" />
*[[Test functions]] in real analysis and distribution theory, which are [[infinitely differentiable function]]s on the real line that are 0 everywhere outside of a given limited [[Interval (mathematics)|interval]]. An example of such a function is the test function, <math display="block">\varphi(t) = \begin{cases}
e^{-1/(1-t^2)}, & -1<t<1, \\
0, & \text{otherwise}.
\end{cases}</math>
*The [[Cantor set]] is a subset of the interval <math>[0,1]</math> that has [[measure (mathematics)|measure]] zero but is [[uncountable]].
*The [[fat Cantor set]] is [[nowhere dense]] but has positive [[measure (mathematics)|measure]].
*The [[Fabius function]] is everywhere [[smoothness|smooth]] but nowhere [[analytic function|analytic]].
*[[Volterra's function]] is [[differentiable function|differentiable]] with [[bounded function|bounded]] derivative everywhere, but the derivative is not [[Riemann-integrable]].
*The Peano [[space-filling curve]] is a continuous [[surjective]] function that maps the unit interval <math>[0,1]</math> onto <math>[0,1]\times[0,1]</math>.
*The [[Dirichlet function]], which is the [[indicator function]] for rationals, is a bounded function that is not [[Riemann integrable]]. 
*The [[Cantor function]] is a [[monotonic]] continuous surjective function that maps <math>[0,1]</math> onto <math>[0,1]</math>, but has zero derivative [[almost everywhere]].
*The [[Minkowski question-mark function]] is continuous and ''strictly'' increasing but has zero derivative almost everywhere.
*Satisfaction classes containing "intuitively false" arithmetical statements can be constructed for [[countable]], recursively saturated [[model theory|models]] of [[Peano arithmetic]]. {{Citation needed|date=April 2018}}
*The [[Osgood curve]] is a [[Jordan curve]] (unlike most [[space-filling curves]]) of positive [[area]].
*An [[exotic sphere]] is [[homeomorphic]] but not [[diffeomorphic]] to the standard Euclidean [[n-sphere]].

At the time of their discovery, each of these was considered highly pathological; today, each has been assimilated into modern mathematical theory. These examples prompt their observers to correct their beliefs or intuitions, and in some cases necessitate a reassessment of foundational definitions and concepts. Over the course of history, they have led to more correct, more precise, and more powerful mathematics. For example, the Dirichlet function is Lebesgue integrable, and convolution with test functions is used to approximate any locally integrable function by smooth functions.<ref group="Note">The approximations converge [[almost everywhere]] and in the [[space of locally integrable functions]].</ref>

Whether a behavior is pathological is by definition subject to personal intuition. Pathologies depend on context, training, and experience, and what is pathological to one researcher may very well be standard behavior to another.

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 that are finite.

Some of the best-known [[paradox]]es, such as [[Banach–Tarski paradox]] and [[Hausdorff paradox]], are based on the existence of [[non-measurable set]]s. Mathematicians, unless they take the minority position of denying the [[axiom of choice]], are in general resigned to living with such sets.{{Citation needed|date=November 2019}}