Editing Weierstrass function (section)

==Riemann function==
The Weierstrass function is based on the earlier Riemann function, claimed to be differentiable nowhere. Occasionally, this function has also been called the Weierstrass function.<ref>{{cite web|first1=Eric W.|last1=Weisstein|url=https://mathworld.wolfram.com/WeierstrassFunction.html|website=[[MathWorld]]|title=Weierstrass Function}}</ref>
<math display="block">
 f(x) = \sum_{n = 1}^\infty \frac{\sin(n^2x)}{n^2}.
</math>

While [[Bernhard Riemann]] strongly claimed that the function is differentiable nowhere, no evidence of this was published by Riemann, and Weierstrass noted that he did not find any evidence of it surviving either in Riemann's papers or orally from his students.

In 1916, [[G. H. Hardy]] confirmed that the function does not have a finite derivative in any value of <math display="inline">\pi x</math> where ''x'' is irrational or is rational with the form of either <math display="inline">\frac{2A}{4B+1}</math> or <math display="inline">\frac{2A+1}{2B}</math>, where ''A'' and ''B'' are integers.<ref name="Hardy"/> In 1969, [[Joseph Gerver]] found that the Riemann function has a defined differential on every value of ''x'' that can be expressed in the form of <math display="inline">\frac{2A+1}{2B+1}\pi</math> with integer ''A'' and ''B'', that is, rational multipliers of <math>\pi</math> with an odd numerator and denominator. On these points, the function has a derivative of <math display="inline">-\frac{1}{2}</math>.<ref>{{Cite journal |title=The Differentiability of the Riemann Function at Certain Rational Multiples of π |first=Joseph |last=Gerver |doi=10.1073/pnas.62.3.668 |journal=Proceedings of the National Academy of Sciences of the United States of America |date=1969 |volume=62 |issue=3 |pages=668–670 |doi-access=free |pmid=16591735 |pmc=223649|bibcode=1969PNAS...62..668G }}</ref> In 1971, J.&nbsp;Gerver showed that the function has no finite differential at the values of ''x'' that can be expressed in the form of <math display="inline">\frac{2A}{2B+1}\pi</math>, completing the problem of the differentiability of the Riemann function.<ref>{{cite journal |title=More on the Differentiability of the Riemann Function |first=Joseph |last=Gerver |doi=10.2307/2373445 |journal=American Journal of Mathematics |date=1971 |volume=93 |issue=1 |pages=33–41 |jstor=2373445 |s2cid=124562827}}</ref>

As the Riemann function is differentiable only on a [[null set]] of points, it is differentiable [[almost nowhere]].