Weierstrass function
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere.
The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points.
[1] Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness.
These types of functions were denounced by contemporaries: Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while Charles Hermite wrote that they were a "lamentable scourge".
The functions were difficult to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of Brownian motion necessitated infinitely jagged functions (nowadays known as fractal curves).
[2] In Weierstrass's original paper, the function was defined as a Fourier series:
This construction, along with the proof that the function is not differentiable over any interval, was first delivered by Weierstrass in a paper presented to the Königliche Akademie der Wissenschaften on 18 July 1872.
[3][4][5] Despite being differentiable nowhere, the function is continuous: Since the terms of the infinite series which defines it are bounded by
, convergence of the sum of the terms is uniform by the Weierstrass M-test with
Since each partial sum is continuous, by the uniform limit theorem, it follows that
Additionally, since each partial sum is uniformly continuous, it follows that
It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be countably infinite or finite.
According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true.
This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points.
Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set (Rademacher's theorem).
Moreover, the fact that the set of non-differentiability points for a monotone function is measure-zero implies that the rapid oscillations of Weierstrass' function are necessary to ensure that it is nowhere-differentiable.
The Weierstrass function was one of the first fractals studied, although this term was not used until much later.
The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line.
Rather between any two points no matter how close, the function will not be monotone.
of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that
[8] The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass's original example.
G. H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions
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
[9] 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
with integer A and B, or rational multipliers of pi with an odd numerator and denominator.
[11] In 1971, J. Gerver showed that the function has no finite differential at the values of x that can be expressed in the form of
, completing the problem of the differentiability of the Riemann function.
It is convenient to write the Weierstrass function equivalently as
is Hölder continuous of exponent α, which is to say that there is a constant C such that
