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Concept
Weierstrass function
Summary
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).
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.
Official source
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