Inmathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes calledpathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes calledwell-behaved ornice. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved.^[1]

A classic example of a pathology is theWeierstrass function, a function that iscontinuous everywhere butdifferentiable nowhere.^[1] The sum of a differentiablefunction and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, using theBaire category theorem, one can show that continuous functions aregenerically nowhere differentiable.^[2]

Such examples were deemed pathological when they were first discovered. To quoteHenri Poincaré:^[3]

Logic sometimes breeds monsters. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. No more continuity, or else continuity but no derivatives, etc. More than this, from the point of view of logic, it is these strange functions that are the most general; those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.

Formerly, when a new function was invented, it was in view of some practical end. To-day they are invented on purpose to show our ancestors' reasonings at fault, and we shall never get anything more than that out of them.

If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages.

— Henri Poincaré, Science and Method (1899), (1914 translation), page 125

Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such asBrownian motion and in applications such as theBlack-Scholes model in finance.

Counterexamples in Analysis is a whole book of such counterexamples.^[4]

Another example of pathological function isDu-Bois Reymondcontinuous function, that can't be represented as aFourier series.^[5]

One famous counterexample in topology is theAlexander horned sphere, showing that topologically embedding the sphereS² inR³ may fail to separate the space cleanly. As a counterexample, it motivated mathematicians to define thetameness property, which suppresses the kind ofwild behavior exhibited by the horned sphere,wild knot, and other similar examples.^[6]

Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same. Yet it does not: it fails to besimply connected.

For the underlying theory, seeJordan–Schönflies theorem.

Counterexamples in Topology is a whole book of such counterexamples.^[7]

Mathematicians (and those in related sciences) very frequently speak of whether amathematical object—afunction, aset, aspace of one sort or another—is"well-behaved". While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, 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 produces aloss of generality of any conclusions reached.

In both pure and applied mathematics (e.g.,optimization,numerical integration,mathematical physics),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 ofcardinality ormeasure) are pathological, but the pathological cases will not arise in practice—unless constructed deliberately.

The term "well-behaved" is generally applied in an absolute sense—either something is well-behaved or it is not. For example:

• Inalgorithmic inference, awell-behaved statistic is monotonic, well-defined, andsufficient.
• InBézout's theorem, twopolynomials are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if theirpolynomial greatest common divisor is a constant.
• Ameromorphic function is a ratio of two well-behaved functions, in the sense of those two functions beingholomorphic.
• TheKarush–Kuhn–Tucker conditions are first-order necessary conditions for a solution in a well-behavednonlinear programming problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied.
• Inprobability, events contained in theprobability space's correspondingsigma-algebra are well-behaved, as aremeasurable functions.

Unusually, the term could also be applied in a comparative sense:

• Incalculus:
  • Analytic functions are better-behaved than generalsmooth functions.
  • Smooth functions are better-behaved than general differentiable functions.
  • Continuousdifferentiable 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 thanRiemann-integrable functions on compact sets.
  • Riemann-integrable functions are better-behaved thanLebesgue-integrable functions.
  • Lebesgue-integrable functions are better-behaved than general functions.
• Intopology:
  • Continuous functions are better-behaved than discontinuous ones.
  • Euclidean space is better-behaved thannon-Euclidean geometry.
  • Attractivefixed points are better-behaved than repulsive fixed points.
  • Hausdorff topologies are better-behaved than those in arbitrarygeneral topology.
  • Borel sets are better-behaved than arbitrarysets ofreal numbers.
  • Spaces withinteger dimension are better-behaved than spaces withfractal dimension.
• Inabstract algebra:
  • Groups are better-behaved thanmagmas andsemigroups.
  • Abelian groups are better-behaved than non-Abelian groups.
  • Finitely-generated Abelian groups are better-behaved than non-finitely-generated Abelian groups.
  • Finite-dimensionalvector spaces are better-behaved thaninfinite-dimensional ones.
  • Fields are better-behaved thanskew fields or generalrings.
  • Separablefield extensions are better-behaved than non-separable ones.
  • Normed division algebras are better-behaved than general composition algebras.

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:

• Ranked-choice voting is commonly described as a pathologicalsocial choice function, because of its tendency to eliminate candidates forwinning too many votes.^[8]
• The discovery ofirrational numbers by the school ofPythagoras in ancient Greece; for example, the length of the diagonal of aunit square, that is${\displaystyle \sqrt{2}}$ .
• The discovery ofcomplex numbers in the 16th century in order to find the roots ofcubic andquarticpolynomial functions.
• Somenumber fields haverings of integers that do not form aunique factorization domain, for example theextended field${\displaystyle \mathbb{Q}(\sqrt{-5})}$ .
• The discovery offractals and other "rough" geometric objects (seeHausdorff dimension).
• Weierstrass function, areal-valued function on thereal line, that iscontinuous everywhere butdifferentiable nowhere.^[1]
• Test functions inreal analysis and distribution theory, which areinfinitely differentiable functions on the real line that are 0 everywhere outside of a given limitedinterval. An example of such a function is the test function, 
• TheCantor set is a subset of the interval${\displaystyle [0,1]}$  that hasmeasure zero but isuncountable.
• Thefat Cantor set isnowhere dense but has positivemeasure.
• TheFabius function is everywheresmooth but nowhereanalytic.
• Volterra's function isdifferentiable withbounded derivative everywhere, but the derivative is notRiemann-integrable.
• The Peanospace-filling curve is a continuoussurjective function that maps the unit interval${\displaystyle [0,1]}$  onto${\displaystyle [0,1]\times [0,1]}$ .
• TheDirichlet function, which is theindicator function for rationals, is a bounded function that is notRiemann integrable.
• TheCantor function is amonotonic continuous surjective function that maps${\displaystyle [0,1]}$  onto${\displaystyle [0,1]}$ , but has zero derivativealmost everywhere.
• TheMinkowski question-mark function is continuous andstrictly increasing but has zero derivative almost everywhere.
• Satisfaction classes containing "intuitively false" arithmetical statements can be constructed forcountable, recursively saturatedmodels ofPeano arithmetic.^{[citation needed]}
• TheOsgood curve is aJordan curve (unlike mostspace-filling curves) of positivearea.
• Anexotic sphere ishomeomorphic but notdiffeomorphic to the standard Euclideann-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 anylocally integrable function by smooth functions.^{[Note 1]}

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, instatistics, theCauchy distribution does not satisfy thecentral limit theorem, even though its symmetricbell-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-knownparadoxes, such asBanach–Tarski paradox andHausdorff paradox, are based on the existence ofnon-measurable sets. Mathematicians, unless they take the minority position of denying theaxiom of choice, are in general resigned to living with such sets.^{[citation needed]}

Incomputer science,pathological has a slightly different sense with regard to the study ofalgorithms. Here, an input (or set of inputs) is said to bepathological if it causes atypical behavior from the algorithm, such as a violation of its average casecomplexity, or even its correctness. For example,hash tables generally have pathological inputs: sets of keys thatcollide on hash values.Quicksort normally has${\displaystyle O(n\log n)}$  time complexity, but deteriorates to${\displaystyle O(n^2)}$  when it is given input that triggers suboptimal behavior.

The term is often used pejoratively, as a way of dismissing such inputs as being specially designed to break a routine that is otherwise sound in practice (compare withByzantine). On the other hand, awareness of pathological inputs is important, as they can be exploited to mount adenial-of-service attack on a computer system. Also, the term in this sense is a matter of subjective judgment as with its other senses. Given enough run time, a sufficiently large and diverse user community (or other factors), an input which may be dismissed as pathological could in fact occur (as seen in thefirst test flight of theAriane 5).

A similar but distinct phenomenon is that ofexceptional objects (andexceptional isomorphisms), which occurs when there are a "small" number of exceptions to a general pattern (such as a finite set of exceptions to an otherwise infinite rule). By contrast, in cases of pathology, often most or almost all instances of a phenomenon are pathological (e.g., almost all real numbers are irrational).

Subjectively, exceptional objects (such as theicosahedron orsporadic simple groups) are generally considered "beautiful", unexpected examples of a theory, while pathological phenomena are often considered "ugly", as the name implies. Accordingly, theories are usually expanded to include exceptional objects. For example, theexceptional Lie algebras are included in the theory ofsemisimple Lie algebras: the axioms are seen as good, the exceptional objects as unexpected but valid.

By contrast, pathological examples are instead taken to point out a shortcoming in the axioms, requiring stronger axioms to rule them out. For example, requiring tameness of an embedding of a sphere in theSchönflies problem. In general, one may study the more general theory, including the pathologies, which may provide its own simplifications (the real numbers have properties very different from the rationals, and likewise continuous maps have very different properties from smooth ones), but also the narrower theory, from which the original examples were drawn.

• Fractal curve
• List of mathematical jargon
• Runge's phenomenon
• Gibbs phenomenon
• Paradoxical set

• Pathological Structures & Fractals – Extract of an article byFreeman Dyson, "Characterising Irregularity", Science, May 1978

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