While the term has no fixed formal definition, it generally refers to the quality of satisfying a list of prevailing conditions, which might be dependent on context, mathematical interests, fashion, and taste. Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object-a function, a set, a space of one sort or another-is "well-behaved". Yet it does not: it fails to be simply connected.įor the underlying theory, see Jordan–Schönflies theorem.Ĭounterexamples in Topology is a whole book of such counterexamples. In this case, the topology of an ever-descending chain of interlocking loops of continuous pieces of the sphere in the limit fully reflects that of the common sphere, and one would expect the outside of it, after an embedding, to work the same. Like many other pathologies, the horned sphere in a sense plays on infinitely fine, recursively generated structure, which in the limit violates ordinary intuition. As a counterexample, it motivated mathematicians to define the tameness property, which suppresses the kind of wild behavior exhibited by the horned sphere, wild knot, and other similar examples. One famous counterexample in topology is the Alexander horned sphere, showing that topologically embedding the sphere S 2 in R 3 may fail to separate the space cleanly. Since Poincaré, nowhere differentiable functions have been shown to appear in basic physical and biological processes such as Brownian motion and in applications such as the Black-Scholes model in finance.Ĭounterexamples in Analysis is a whole book of such counterexamples. Henri Poincaré, Science and Method (1899), (1914 translation), page 125 If you don't do so, the logicians might say, you will only reach exactness by stages. He would have to set the beginner to wrestle with this collection of monstrosities. If logic were the teacher's only guide, he would have to begin with the most general, that is to say, with the most weird, functions. To-day they are invented on purpose to show our ancestors' reasonings at fault, and we shall never get anything more than that out of them. More than this, from the point of view of logic, it is these strange functions that are the most general those that are met without being looked for no longer appear as more than a particular case, and they have only quite a little corner left them.įormerly, when a new function was invented, it was in view of some practical end. No more continuity, or else continuity but no derivatives, etc. For half a century there has been springing up a host of weird functions, which seem to strive to have as little resemblance as possible to honest functions that are of some use. Such examples were deemed pathological when they were first discovered: In fact, using the Baire category theorem, one can show that continuous functions are generically nowhere differentiable. 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 analysis Ī classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. These terms are sometimes useful in mathematical research and teaching, but there is no strict mathematical definition of pathological or well-behaved. On the other hand, if a phenomenon does not run counter to intuition, In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. The Weierstrass function is continuous everywhere but differentiable nowhere.
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