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.
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:
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.
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.
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