Editing Pathological (mathematics) (section)

==In analysis==
A classic example of a pathology is the [[Weierstrass function]], a function that is [[Continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere.<ref name=":1" /> The sum of a differentiable [[Function (mathematics)|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, using the [[Baire category theorem]], one can show that continuous functions are [[Generic property|generically]] nowhere differentiable.<ref>{{Cite web|url=https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one|title=Baire Category & Nowhere Differentiable Functions (Part One)|website=www.math3ma.com|access-date=2019-11-29}}</ref>

Such examples were deemed pathological when they were first discovered. To quote [[Henri Poincaré]]:<ref>{{Cite book |author=Kline, Morris |url=http://worldcat.org/oclc/1243569759 |title=Mathematical thought from ancient to modern times. |date=1990 |publisher=Oxford University Press |pages=973 |oclc=1243569759}}</ref>

{{blockquote|Logic sometimes breeds monsters. 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. No more continuity, or else continuity but no derivatives, etc. 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.

Formerly, when a new function was invented, it was in view of some practical end. 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.

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. He would have to set the beginner to wrestle with this collection of monstrosities. If you don't do so, the logicians might say, you will only reach exactness by stages.|[[Henri Poincaré]]|Science and Method (1899)|source=(1914 translation), page 125}}

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.

''Counterexamples in Analysis'' is a whole book of such counterexamples.<ref>{{Cite book |last=Gelbaum |first=Bernard R. |url=https://www.worldcat.org/oclc/527671 |title=Counterexamples in analysis |date=1964 |publisher=Holden-Day |others=John M. H. Olmsted |isbn=0-486-42875-3 |location=San Francisco |oclc=527671}}</ref>

Another example of pathological function is [[Paul du Bois-Reymond|Du-Bois Reymond]] [[continuous function]], that can't be represented as a [[Fourier series]].<ref>{{Cite book |last=Jahnke |first=Hans Niels |title=A history of analysis |date=2003 |publisher=American mathematical society |isbn=978-0-8218-2623-2 |series=History of mathematics |location=Providence (R.I.) |pages=187}}</ref>