Pathological

The term "pathological" is used in mathematics to refer to an example specifically cooked up to violate certain almost universally valid properties. Pathological problems often provide interesting examples of counterintuitive behavior, as well as serving as an excellent illustration of why very detailed conditions of applicability are required in order for many mathematical statements to be universally true.

For example, the pathological Weierstrass and Blancmange functions are examples of a continuous function that is nowhere differentiable, a possibility that many students of calculus find quite surprising.

In 1899, Poincaré remarked on the proliferation of pathological functions, "Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.

In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum" (Kleiner 1989; Kline 1990, p. 973).

REFERENCES:

Kleiner, I. "Evolution of the Function Concept: A Brief Survey." Coll. Math. J. 20, 282-300, 1989.

Kline, M. Mathematical Thought from Ancient to Modern Times. Oxford, England: Oxford University Press, 1990.