Well behaved

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

A classic example of a pathological structure 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, the majority of functions are nowhere differentiable, and relatively few can ever be described or studied. In general most useful functions also have some sort of physical basis or practical application, which means that they cannot be pathological at the level of hard mathematics or logic; absent certain limiting cases like the delta distribution, they tend to be quite well-behaved and intuitive. 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, context-dependent, and subject to wearing off. Its meaning in any particular case resides in the community of mathematicians, and not necessarily within mathematics itself. Also, the quotation shows how mathematics often progresses via counter-examples to what is thought intuitive or expected; for instance, the "lack of derivatives" mentioned is intimately connected with current study of magnetic reconnection events in solar plasma.

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Wikipedia