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Function that is continuous everywhere but differentiable nowhere
Plot of Weierstrass function over the interval [−2, 2]. Like other fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the global plot.
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 impossible to visualize until the arrival of computers in the next century, so the proof of the result relied entirely on technically demanding theoretical steps. 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).
Animation based on the increasing of the b value from 0.1 to 5.
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 "small" in some sense. 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 dimensionD of the graph of the classical Weierstrass function was an open problem until 2018: while it was generally believed that D is 2 + logba, only after more than 30 years[clarification needed] 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.
It is convenient to write the Weierstrass function equivalently as
for . Then W?(x) is Hölder continuous of exponent ?, which is to say that there is a constant C such that
In a measure-theoretic sense: when the space C([0, 1]; R) is equipped with classical Wiener measure?, the collection of functions that are differentiable at even a single point of [0, 1] has ?-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).
^At least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes.
Around 1831, Bernard Bolzano (1781 - 1848), a Czech mathematician, philosopher, and Catholic priest, constructed such a function; however, it was not published until 1922. See:
Martin Ja?ek (1922) "Funkce Bolzanova" (Bolzano's function), ?asopis pro P?stování Matematiky a Fyziky (Journal for the Cultivation of Mathematics and Physics), vol. 51, no. 2, pages 69-76 (in Czech and German).
Karel Rychlík (1923) "Über eine Funktion aus Bolzanos handschriftlichem Nachlasse" (On a function from Bolzano's literary remains in manuscript), Sitzungsberichte der königlichen Böhmischen Gesellschaft der Wissenschaften (Prag) (Proceedings of the Royal Bohemian Society of Philosophy in Prague) (for the years 1921-1922), Class II, no. 4, pages 1-20. (Sitzungsberichte was continued as: V?stník Královské ?eské spole?nosti nauk, t?ída matematicko-p?írodov?decká (Journal of the Royal Czech Society of Science, Mathematics and Natural Sciences Class).)
Around 1860, Charles Cellérier (1818 - 1889), a professor of mathematics, mechanics, astronomy, and physical geography at the University of Geneva, Switzerland, independently formulated a continuous, nowhere differentiable function that closely resembles Weierstrass's function. Cellérier's discovery was, however, published posthumously:
^On page 560 of the 1872 Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (Monthly Reports of the Royal Prussian Academy of Science in Berlin), there is a brief mention that on 18 July, "Hr. Weierstrass las über stetige Funktionen ohne bestimmte Differentialquotienten" (Mr. Weierstrass read [a paper] about continuous functions without definite [i.e., well-defined] derivatives [to members of the Academy]). However, Weierstrass's paper was not published in the Monatsberichte.
David, Claire (2018), "Bypassing dynamical systems : A simple way to get the box-counting dimension of the graph of the Weierstrass function", Proceedings of the International Geometry Center, Academy of Sciences of Ukraine, 11 (2): 53-68, doi:10.15673/tmgc.v11i2.1028
Weierstrass, Karl (18 July 1872), Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen, Königlich Preussische Akademie der Wissenschaften
English translation: Edgar, Gerald A. (1993), "On continuous functions of a real argument that do not possess a well-defined derivative for any value of their argument", Classics on Fractals, Studies in Nonlinearity, Addison-Wesley Publishing Company, pp. 3-9, ISBN978-0-201-58701-2