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.
In analysis
A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. 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, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable.
Such examples were deemed pathological when they were first discovered: To quote Henri Poincaré:
Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such as Brownian motion and in applications such as the Black-Scholes model in finance.
Counterexamples in Analysis is a whole book of such counterexamples.
In topology
One famous counterexample in topology is the Alexander horned sphere, showing that topologically embedding the sphere S2 in R3 may fail to separate the space cleanly. As a counterexample, it motivated mathematicians to define the tameness property, which suppresses the kind of wild behavior exhibited by the horned sphere, wild knot, and other similar examples.
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 be simply connected.
For the underlying theory, see Jordan–Schönflies theorem.
Counterexamples in Topology is a whole book of such counterexamples.
Well-behaved
Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object—a function, a set, a space 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 a loss 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 of cardinality or measure) 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:
In algorithmic inference, a well-behaved statistic is monotonic, well-defined, and sufficient. In Bézout's theorem, two polynomials are well-behaved, and thus the formula given by the theorem for the number of their intersections is valid, if their polynomial greatest common divisor is a constant. A meromorphic function is a ratio of two well-behaved functions, in the sense of those two functions being holomorphic. The Karush–Kuhn–Tucker conditions are first-order necessary conditions for a solution in a well-behaved nonlinear programming problem to be optimal; a problem is referred to as well-behaved if some regularity conditions are satisfied. In probability, events contained in the probability space's corresponding sigma-algebra are well-behaved, as are measurable functions.
Unusually, the term could also be applied in a comparative sense: