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 infinity of possible functions, relatively few will ever be studied by mathematicians, and those that do come to their attention as being interesting or useful will tend 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. 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. |
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—Henri Poincaré, 1899 |
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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.
Read more about this topic: Pathological (mathematics)
Famous quotes containing the words pathological and/or functions:
“Analysis does not set out to make pathological reactions impossible, but to give the patient’s ego freedom to decide one way or another.”
—Sigmund Freud (1856–1939)
“The mind is a finer body, and resumes its functions of feeding, digesting, absorbing, excluding, and generating, in a new and ethereal element. Here, in the brain, is all the process of alimentation repeated, in the acquiring, comparing, digesting, and assimilating of experience. Here again is the mystery of generation repeated.”
—Ralph Waldo Emerson (1803–1882)