In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

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. 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).

Construction

In Weierstrass's original paper, the function was defined as a Fourier series:

where ,   is a positive odd integer, and

The minimum value of  for which there exists  such that these constraints are satisfied is . 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.

Despite never being differentiable, the function is continuous: Since the terms of the infinite series which defines it are bounded by ±an and this has finite sum for 0 < a < 1, convergence of the sum of the terms is uniform by the Weierstrass M-test with Mn = an. Since each partial sum is continuous, by the uniform limit theorem, it follows that f is continuous. Additionally, since each partial sum is uniformly continuous, it follows that f is also uniformly continuous.

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). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved.

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.

The computation of the Hausdorff dimension D of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that D = . That D is strictly less than 2 follows from the conditions on  and  from above. Only after more than 30 years was this proved rigorously.

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. For example, the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions 0 < a < 1, ab ≥ 1.

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.

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 doesn't have a finite derivative in any value of  where x is irrational or is rational with the form of either  or , where A and B are integers. 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. On these points, the function has a derivative of . 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.

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

Hölder continuity
It is convenient to write the Weierstrass function equivalently as

for .  Then Wα(x) is Hölder continuous of exponent α, which is to say that there is a constant C such that

for all x and y. Moreover, W1 is Hölder continuous of all orders  but not Lipschitz continuous.

Density of nowhere-differentiable functions

It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions:
 In a topological sense: the set of nowhere-differentiable real-valued functions on [0, 1] is comeager in the vector space C([0, 1]; R) of all continuous real-valued functions on [0, 1] with the topology of uniform convergence.
 In a measure-theoretic sense: when the space C([0, 1]; R) is equipped with classical Wiener measure γ, the collection of functions that are differentiable at even a single point of [0, 1] has γ-measure zero. The same is true even if one takes finite-dimensional "slices" of C([0, 1]; R), in the sense that the nowhere-differentiable functions form a prevalent subset of C([0, 1]; R).

See also
Blancmange curve
Koch snowflake
Nowhere continuous function

Notes

References 

English translation:

External links 
  (a different Weierstrass Function which is also continuous and nowhere differentiable)
 Nowhere differentiable continuous function proof of existence using Banach's contraction principle.
 Nowhere monotonic continuous function proof of existence using the Baire category theorem.
 
 Weierstrass function in the complex plane Beautiful fractal.
 SpringerLink - Journal of Fourier Analysis and Applications, Volume 16, Number 1 Simple Proofs of Nowhere-Differentiability for Weierstrass's Function and Cases of Slow Growth
 Weierstrass functions: continuous but not differentiable anywhere
 The Weierstrass Function by Brent Nelson at Berkeley, showing non differentiable

Theory of continuous functions
Measure theory
Real analysis
Fractal curves
Types of functions
수학에서 바이어슈트라스 함수(-函數, )는 칸토어 함수의 한 예이다. 모든 점에서 연속이나, 모든 점에서 미분 불능이다. 독일의 수학자 카를 바이어슈트라스가 제안하였다.

바이어슈트라스 함수는 '모든 연속함수는 많아 봐야 고립점들의 집합에서만 미분 불가능'이라는 생각의 처음 출간된 (1872) 반례이기에 역사적 중요성을 띤다.

정의 

아래는 바이어슈트라스의 논문 원본에서의 정의이다.

(단, , 는 양의 홀수, )

위 조건을 만족하는 의 최솟값은 이다. 이 함수의 정의와 모든 점에서의 미분 불가능성의 증명은 1872년 7월 18일 바이어슈트라스가 프로이센 과학 아카데미에 제출한 논문에서 처음 발표되었다.

실해석학에서 바이어슈트라스 함수는 흔히 그와 비슷한 정의와 성질을 가진 함수를 지칭하는 데 쓰인다. 그 예로 무한급수 안의 코사인 함수를 삼각파로 대신한 것이 있다.

, 의 전제 하에 모든 점에서 미분 불능임을 고드프리 해럴드 하디가 증명하였다.

연속성과 미분가능성 
바이어슈트라스 함수는 모든 점에서 연속이나, 모든 점에서 미분 불능이다.

모든 점에서의 연속성은 바이어슈트라스 M-판정법에서 을 취하면 쉽게 증명된다.

모든 점에서의 미분 불가능성을 증명하려면, 임의의 가 주어질 때, 두 수열 을 구성하여 아래를 증명하면 된다.

위에서 와 는 각각 하극한과 상극한이다.

모든 연속함수는 모든 점에서 미분 가능하거나 아주 작은 집합에서만 미분 불가능이라는 관념은 오래 존재해왔다. 바이어슈트라스의 논문에 의하면, 가우스를 비롯한 초기 수학자들은 이를 자주 기정 사실로 취급하였다. 반례를 찾아내기가 어려운 것이 이유일 수 있다. 실제로 일부 더 강한 조건의 연속성(예를 들어 립시츠 연속성)은 거의 어디서나 미분 가능임을 함의한다(레이드매처의 정리). 우리가 연속 함수를 그릴 때에도 결과물은 대개 립시츠 연속성 등의 더 좋은 성질을 가진다.

프랙탈 
바이어슈트라스 함수는 최초의 프랙탈 중 하나로, 자기 유사성을 지닌다. 따라서 곡선을 확대해도 직선에 가까워지지 않으며 임의의 두 점 사이에서 단조성이 없다. 바이어슈트라스 함수의 하우스도르프 차원 는 를 상계로 갖는다. 일반적으로 라고 추측하나 아직 증명되지 않았다.

횔더 연속성 
다음은 바이어슈트라스 함수의 동등한 표현이다.

이 중 는 을 만족한다. 는 α-횔더 연속성을 가진다. 즉, 상수 가 존재하여 임의의 에 대하여

더 나아가 는 임의의 에 대해 α-횔더 연속이지만 립시츠 연속(즉 1-횔더 연속)은 아니다.

모든 곳에서 미분 불능인 함수의 조밀성 
바이어슈트라스 함수가 소수 특례인 것은 아니다. 병적이지만 연속함수에게는 일반적이라는 것이다.

 위상수학의 관점에서,  위의 연속 실가 함수의 균등수렴 위상을 가진 벡터 공간 에서,  위의 온 곳에서 미분 불능인 함수의 집합은 나머지 집합을 이룬다.
 측도론의 관점에서, 공간 에 고전적 위너 측도 를 부여했을 때 적어도 한 점에서 미분 가능한 함수의 집합은 γ-영측도이다.

같이 보기 
 코크 곡선
 블랑망제 곡선
 바이어슈트라스 시그마 함수
 바이어슈트라스 제타 함수

각주

참고 문헌 
 B.R. Gelbaum and J.M.H. Olmstead, Counterexamples in Analysis, Holden Day Publisher (June 1964).
 Karl Weierstrass, "Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen,"  in: Königlich Preussichen Akademie der Wissenschaften, Mathematische Werke von Karl Weierstrass (Berlin, Germany: Mayer & Mueller, 1895), vol. 2, pages 71–74.; English translation: On continuous functions of a real argument that do not possess a well-defined derivative for any value of their argument, in: G.A. Edgar, Classics on Fractals, Addison-Wesley Publishing Company, 1993, 3–9.
 G.H. Hardy, "Weierstrass's nondifferentiable function," Trans. Amer. Math. Soc., 17(1916), 301–325.
 K. Falconer, The Geometry of Fractal Sets, Oxford (1984).

외부 링크 
 
 Nowhere differentiable continuous function  바나흐 고정점 정리에 의한 존재성 증명.
 Nowhere monotonic continuous function  베르 범주 정리에 의한 존재성 증명.
 
 Weierstrass function in the complex plane   아름다운 프랙탈.
 SpringerLink - Journal of Fourier Analysis and Applications, Volume 16, Number 1  Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth
 Weierstrass functions: continuous but not differentiable anywhere 

연속 함수
측도론
실해석학
프랙탈 곡선
함수의 종류