Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved. A notable 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.

Pathological functions

A classic example is the Weierstrass function, which 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, 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 great number of possible functions; the majority are nowhere differentiable, and relatively few can ever be described and studied, of which most that are interesting or useful also turn out to be well-behaved. To quote Henri Poincaré: