In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive.

Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S² in R³ may fail to "separate the space cleanly", unless an extra condition of tameness is used to suppress possible wild behaviour. See Jordan-Schönflies theorem.

One can therefore say that (particularly in mathematical analysis and set theory) those searching for the "pathological" are like experimentalists, interested in knocking down potential theorems, in contrast to finding general statements widely applicable. Each activity has its role within mathematics.

Pathological functions

A classic example is the Weierstrass function, which is continuous everywhere but differentiable nowhere. Obviously, the sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions. In fact, by the Baire category theorem one can show that continuous functions are typically or generically nowhere differentiable.

In layman's terms, this is because of the vast infinity of possible functions, relatively few will ever be studied by mathematicians, and those that do come to their attention as being interesting or useful will tend to be well-behaved. To quote Henri Poincaré:

This highlights the fact that the term pathological is subjective or at least context-dependent, and its meaning in any particular case resides in the community of mathematicians, not necessarily within the subject matter of mathematics itself.

Pathological examples

Pathological examples often have some undesirable or unusual properties that make it difficult to contain or explain within a theory. Such pathological behaviour often prompts new investigation which leads to new theory and more general results. For example, three important historical examples of this are the following:

• The discovery of irrational numbers by the school of Pythagoras in ancient Greece; the first example of an irrational number they found was the length of the diagonal of a unit square, that is $\backslash sqrt\{2\}$

• The discovery of number fields whose rings of integers do not form a unique factorization domain, for example the field $\backslash mathbb\{Q\}(\backslash sqrt\{-5\})$.

• The discovery of fractals and other "rough" geometric objects (see Hausdorff dimension).

• Weierstrass function, a real valued function on the real line, that is continuous everywhere but differentiable nowhere.

• In Fourier analysis, test functions are complex valued functions on the real line, that are 0 everywhere outside of a limited interval (hence all derivatives will also be 0 outside of that interval) and $\backslash neq\; 0$ inside of the interval, but are still infinitely differentiable everywhere.

At the time of their discovery, each of these were considered highly pathological; today, each has been assimilated, which is to say, explained by an extensive general theory.

Again, to reiterate, it should be pointed out that such judgments about what is or is not pathological are inherently subjective or at least vary with context and depend on both training and experience — what is pathological to one researcher may very well be standard behaviour 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 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.

Computer science

In computer science, pathological has a slightly different sense with regard to the study of algorithms. Here, an input (or set of inputs) is said to be pathological if it causes atypical behavior from the algorithm, such as a violation of its average case complexity, or even its correctness. For example, hash tables generally have pathological inputs: sets of keys that collide on hash values. Quicksort normally has O(n log n) time complexity, but deteriorates to O(n²) when given input that triggers suboptimal behaviour.

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 with Byzantine). On the other hand, awareness of pathological inputs is important as they can be exploited to mount a denial-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 the first test flight of the Ariane 5).

See also

• Well-behaved